\(\int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx\) [251]

Optimal result

Integrand size = 24, antiderivative size = 794 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\frac {\sqrt {a+b x^2+c x^4}}{2 e}-\frac {\sqrt {c} d x \sqrt {a+b x^2+c x^4}}{e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {c d^4+b d^2 e^2+a e^4} \text {arctanh}\left (\frac {\sqrt {c d^4+b d^2 e^2+a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 e^3}+\frac {\left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} e^3}-\frac {\sqrt {c d^4+b d^2 e^2+a e^4} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 e^3}+\frac {\sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{e^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} d \left (2 c d^2+b e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}} \] Output:

1/2*(c*x^4+b*x^2+a)^(1/2)/e-c^(1/2)*d*x*(c*x^4+b*x^2+a)^(1/2)/e^2/(a^(1/2) 
+c^(1/2)*x^2)+1/2*(a*e^4+b*d^2*e^2+c*d^4)^(1/2)*arctanh((a*e^4+b*d^2*e^2+c 
*d^4)^(1/2)*x/d/e/(c*x^4+b*x^2+a)^(1/2))/e^3+1/4*(b*e^2+2*c*d^2)*arctanh(1 
/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(1/2)/e^3-1/2*(a*e^4+b*d^2 
*e^2+c*d^4)^(1/2)*arctanh(1/2*(b*d^2+2*a*e^2+(b*e^2+2*c*d^2)*x^2)/(a*e^4+b 
*d^2*e^2+c*d^4)^(1/2)/(c*x^4+b*x^2+a)^(1/2))/e^3+a^(1/4)*c^(1/4)*d*(a^(1/2 
)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(s 
in(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/e^2/(c*x^ 
4+b*x^2+a)^(1/2)-1/2*a^(1/4)*d*(b*e^2+2*c*d^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x 
^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4 
)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(1/4)/e^2/(c^(1/2)*d^2+a^( 
1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)-1/4*(c^(1/2)*d^2-a^(1/2)*e^2)*(a*e^4+b*d^2 
*e^2+c*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2 
)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(c^(1/2)*d^2+a^(1/ 
2)*e^2)^2/a^(1/2)/c^(1/2)/d^2/e^2,1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(1/4) 
/c^(1/4)/d/e^4/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.47 (sec) , antiderivative size = 5994, normalized size of antiderivative = 7.55 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\text {Result too large to show} \] Input:

Integrate[Sqrt[a + b*x^2 + c*x^4]/(d + e*x),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.80 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {2266, 1523, 27, 1511, 27, 1416, 1509, 1576, 1162, 1269, 1092, 219, 1154, 219, 2222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx\)

\(\Big \downarrow \) 2266

\(\displaystyle d \int \frac {\sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 1523

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\int \frac {c d^2+b e^2-\sqrt {a} \sqrt {c} e^2+c \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\int \frac {c d^2+b e^2-\sqrt {a} \sqrt {c} e^2+c \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 1511

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\left (b e^2+2 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\left (b e^2+2 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 1416

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 1509

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx\)

\(\Big \downarrow \) 1576

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \int \frac {\sqrt {c x^4+b x^2+a}}{d^2-e^2 x^2}dx^2\)

\(\Big \downarrow \) 1162

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\int \frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{2 e^2}-\frac {\sqrt {a+b x^2+c x^4}}{e^2}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {2 \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e^2}-\left (b+\frac {2 c d^2}{e^2}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{2 e^2}-\frac {\sqrt {a+b x^2+c x^4}}{e^2}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {2 \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e^2}-2 \left (b+\frac {2 c d^2}{e^2}\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{2 e^2}-\frac {\sqrt {a+b x^2+c x^4}}{e^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {2 \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e^2}-\frac {\left (b+\frac {2 c d^2}{e^2}\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}}{2 e^2}-\frac {\sqrt {a+b x^2+c x^4}}{e^2}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {-\frac {4 \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{4 \left (c d^4+b e^2 d^2+a e^4\right )-x^4}d\left (-\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (b+\frac {2 c d^2}{e^2}\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}}{2 e^2}-\frac {\sqrt {a+b x^2+c x^4}}{e^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle d \left (\frac {\left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (b e^2+2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {2 \sqrt {a e^4+b d^2 e^2+c d^4} \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{e^2}-\frac {\left (b+\frac {2 c d^2}{e^2}\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}}{2 e^2}-\frac {\sqrt {a+b x^2+c x^4}}{e^2}\right )\)

\(\Big \downarrow \) 2222

\(\displaystyle d \left (\frac {\left (c d^4+b e^2 d^2+a e^4\right ) \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d^4+b e^2 d^2+a e^4} x}{d e \sqrt {c x^4+b x^2+a}}\right )}{2 d e \sqrt {c d^4+b e^2 d^2+a e^4}}+\frac {\left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\frac {\left (2 c d^2+b e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {\sqrt {c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {x \sqrt {c x^4+b x^2+a}}{\sqrt {c} x^2+\sqrt {a}}\right )}{\sqrt {a}}}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {2 \sqrt {c d^4+b e^2 d^2+a e^4} \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{e^2}-\frac {\left (\frac {2 c d^2}{e^2}+b\right ) \text {arctanh}\left (\frac {2 c x^2+b}{2 \sqrt {c} \sqrt {c x^4+b x^2+a}}\right )}{\sqrt {c}}}{2 e^2}-\frac {\sqrt {c x^4+b x^2+a}}{e^2}\right )\)

Input:

Int[Sqrt[a + b*x^2 + c*x^4]/(d + e*x),x]
 

Output:

-1/2*(e*(-(Sqrt[a + b*x^2 + c*x^4]/e^2) + (-(((b + (2*c*d^2)/e^2)*ArcTanh[ 
(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c]) + (2*Sqrt[c*d 
^4 + b*d^2*e^2 + a*e^4]*ArcTanh[(b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/ 
(2*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]*Sqrt[a + b*x^2 + c*x^4])])/e^2)/(2*e^2) 
)) + d*(-((-((Sqrt[c]*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(-((x*Sqrt[a + b*x^2 + c 
*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a 
 + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x 
)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4] 
)))/Sqrt[a]) + ((2*c*d^2 + b*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 
+ c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4) 
], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*x^4 
]))/(e^2*((Sqrt[c]*d^2)/Sqrt[a] + e^2))) + ((c*d^4 + b*d^2*e^2 + a*e^4)*(( 
(Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTanh[(Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]*x)/(d 
*e*Sqrt[a + b*x^2 + c*x^4])])/(2*d*e*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]) + (( 
Sqrt[a]/d^2 - Sqrt[c]/e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4 
)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*S 
qrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sq 
rt[c]))/4])/(4*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/(Sqrt[a]*e^2*((S 
qrt[c]*d^2)/Sqrt[a] + e^2)))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1523

Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(c*d^2 - b*d*e + a*e^2)/(e*(e - d*q))   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Simp[1/(e*(e 
 - d*q))   Int[(c*d - b*e + a*e*q - (c*e - a*d*q^3)*x^2)/Sqrt[a + b*x^2 + c 
*x^4], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c 
*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 2266

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbo 
l] :> Simp[d   Int[(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Simp[e 
 Int[x*((a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2)), x], x] /; FreeQ[{a, b, c, d 
, e}, x] && IntegerQ[p + 1/2]
 

Maple [A] (verified)

Time = 5.06 (sec) , antiderivative size = 755, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e}-\frac {d \left (b \,e^{2}+c \,d^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e^{4} \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (\frac {b \,e^{2}+c \,d^{2}}{e^{3}}-\frac {b}{2 e}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {c d a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 e^{2} \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {\left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{5}}\)	\(755\)
elliptic	\(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e}-\frac {d \left (b \,e^{2}+c \,d^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e^{4} \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (\frac {b \,e^{2}+c \,d^{2}}{e^{3}}-\frac {b}{2 e}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {c d a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 e^{2} \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {\left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{5}}\)	\(755\)
risch	\(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e}-\frac {\frac {-\frac {e \left (b \,e^{2}+2 c \,d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {c \,d^{3} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b d \,e^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {c d \,e^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}}{e^{3}}-\frac {2 \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{4}}}{2 e}\)	\(894\)

Input:

int((c*x^4+b*x^2+a)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/2*(c*x^4+b*x^2+a)^(1/2)/e-1/4*d*(b*e^2+c*d^2)/e^4*2^(1/2)/((-b+(-4*a*c+b 
^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4* 
a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2) 
*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c) 
^(1/2))+1/2*(1/e^3*(b*e^2+c*d^2)-1/2/e*b)*ln((2*c*x^2+b)/c^(1/2)+2*(c*x^4+ 
b*x^2+a)^(1/2))/c^(1/2)+1/2*c*d/e^2*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^ 
(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2) 
)/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2 
*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^( 
1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2 
),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))+(a*e^4+b*d^2*e^2+c*d^4)/ 
e^5*(-1/2/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+b*d^2 
/e^2+b*x^2+2*a)/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/ 
2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)/d*e*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a* 
x^2)^(1/2)*(1+1/2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2 
)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),2/(-b+(-4*a*c 
+b^2)^(1/2))*a/d^2*e^2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+ 
(-4*a*c+b^2)^(1/2))/a)^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\text {Timed out} \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/(e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{d + e x}\, dx \] Input:

integrate((c*x**4+b*x**2+a)**(1/2)/(e*x+d),x)
 

Output:

Integral(sqrt(a + b*x**2 + c*x**4)/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a}}{e x + d} \,d x } \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/(e*x+d),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)/(e*x + d), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a}}{e x + d} \,d x } \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/(e*x+d),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{d+e\,x} \,d x \] Input:

int((a + b*x^2 + c*x^4)^(1/2)/(d + e*x),x)
 

Output:

int((a + b*x^2 + c*x^4)^(1/2)/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{d+e x} \, dx=\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{e x +d}d x \] Input:

int((c*x^4+b*x^2+a)^(1/2)/(e*x+d),x)
                                                                                    
                                                                                    
 

Output:

int((c*x^4+b*x^2+a)^(1/2)/(e*x+d),x)