\(\int (d+e x) (a+b x^2+c x^4)^{3/2} \, dx\) [256]

Optimal result

Integrand size = 22, antiderivative size = 512 \[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {2 b \left (b^2-8 a c\right ) d x \sqrt {a+b x^2+c x^4}}{35 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b^2-4 a c\right ) e \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{128 c^2}+\frac {d x \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 c}+\frac {1}{7} d x \left (a+b x^2+c x^4\right )^{3/2}+\frac {e \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 c}+\frac {3 \left (b^2-4 a c\right )^2 e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{256 c^{5/2}}+\frac {2 \sqrt [4]{a} b \left (b^2-8 a c\right ) d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 c^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 c^{5/4} \sqrt {a+b x^2+c x^4}} \] Output:

-2/35*b*(-8*a*c+b^2)*d*x*(c*x^4+b*x^2+a)^(1/2)/c^(3/2)/(a^(1/2)+c^(1/2)*x^ 
2)-3/128*(-4*a*c+b^2)*e*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^2+1/35*d*x*(3* 
b*c*x^2+10*a*c+b^2)*(c*x^4+b*x^2+a)^(1/2)/c+1/7*d*x*(c*x^4+b*x^2+a)^(3/2)+ 
1/16*e*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(3/2)/c+3/256*(-4*a*c+b^2)^2*e*arctanh( 
1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(5/2)+2/35*a^(1/4)*b*(-8* 
a*c+b^2)*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(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2) 
)^(1/2))/c^(7/4)/(c*x^4+b*x^2+a)^(1/2)-1/70*a^(1/4)*(a^(1/2)*(-20*a*c+b^2) 
+2*b*(-8*a*c+b^2)/c^(1/2))*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)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*( 
2-b/a^(1/2)/c^(1/2))^(1/2))/c^(5/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.79 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.26 \[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {-128 i \sqrt {2} b \sqrt {c} \left (b^2-8 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) d \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+128 i \sqrt {2} \sqrt {c} \left (-b^4+9 a b^2 c-20 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) d \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (-105 b^3 e+2 b^2 c x (64 d+35 e x)+40 c^2 x \left (2 c x^4 (8 d+7 e x)+a (48 d+35 e x)\right )+4 b c \left (175 a e+2 c x^3 (128 d+105 e x)\right )\right )-105 \left (b^2-4 a c\right )^2 e \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{8960 c^{5/2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[(d + e*x)*(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

((-128*I)*Sqrt[2]*b*Sqrt[c]*(b^2 - 8*a*c)*(-b + Sqrt[b^2 - 4*a*c])*d*Sqrt[ 
(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[ 
b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[ 
2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b 
^2 - 4*a*c])] + (128*I)*Sqrt[2]*Sqrt[c]*(-b^4 + 9*a*b^2*c - 20*a^2*c^2 + b 
^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*d*Sqrt[(b - Sqrt[b^2 - 4 
*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2* 
c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sq 
rt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + S 
qrt[c/(b + Sqrt[b^2 - 4*a*c])]*(2*Sqrt[c]*(a + b*x^2 + c*x^4)*(-105*b^3*e 
+ 2*b^2*c*x*(64*d + 35*e*x) + 40*c^2*x*(2*c*x^4*(8*d + 7*e*x) + a*(48*d + 
35*e*x)) + 4*b*c*(175*a*e + 2*c*x^3*(128*d + 105*e*x))) - 105*(b^2 - 4*a*c 
)^2*e*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + 
 c*x^4]]))/(8960*c^(5/2)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + 
c*x^4])
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2202, 27, 1404, 1432, 1087, 1087, 1092, 219, 1490, 25, 1511, 27, 1416, 1509}

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.

\(\displaystyle \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int d \left (c x^4+b x^2+a\right )^{3/2}dx+\int e x \left (c x^4+b x^2+a\right )^{3/2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle d \int \left (c x^4+b x^2+a\right )^{3/2}dx+e \int x \left (c x^4+b x^2+a\right )^{3/2}dx\)

\(\Big \downarrow \) 1404

\(\displaystyle d \left (\frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+e \int x \left (c x^4+b x^2+a\right )^{3/2}dx\)

\(\Big \downarrow \) 1432

\(\displaystyle d \left (\frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \int \left (c x^4+b x^2+a\right )^{3/2}dx^2\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )+d \left (\frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )+d \left (\frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )+d \left (\frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle d \left (\frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

\(\Big \downarrow \) 1490

\(\displaystyle d \left (\frac {3}{7} \left (\frac {\int -\frac {2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (10 a c+b^2+3 b c x^2\right )}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle d \left (\frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\int \frac {2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

\(\Big \downarrow \) 1511

\(\displaystyle d \left (\frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \sqrt {a} b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

\(\Big \downarrow \) 1416

\(\displaystyle d \left (\frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

\(\Big \downarrow \) 1509

\(\displaystyle d \left (\frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 b \left (b^2-8 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\right )+\frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\)

Input:

Int[(d + e*x)*(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

(e*(((b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*((( 
b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 
2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))))/(16*c)))/2 + 
d*((x*(a + b*x^2 + c*x^4)^(3/2))/7 + (3*((x*(b^2 + 10*a*c + 3*b*c*x^2)*Sqr 
t[a + b*x^2 + c*x^4])/(15*c) - ((-2*b*(b^2 - 8*a*c)*(-((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[c] + (a^(1/4)*(Sqrt[a]*(b^2 - 20*a*c) + (2*b*(b^2 - 8*a*c))/Sqr 
t[c])*(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*c^(1/4)*Sqrt[a + b*x^2 + c*x^4]))/(15*c)))/7)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1404

Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b 
*x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1))   Int[(2*a + b*x^2)*( 
a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* 
c, 0] && GtQ[p, 0] && IntegerQ[2*p]
 

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 1432

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

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), 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] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

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 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.35

method	result	size
elliptic	\(\frac {c e \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8}+\frac {c d \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7}+\frac {3 e b \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16}+\frac {8 b d \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {\left (\frac {5}{4} a c e +\frac {1}{16} b^{2} e \right ) x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (\frac {9}{7} a c d +\frac {3}{35} d \,b^{2}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (\frac {5 a b e}{4}-\frac {3 b \left (\frac {5}{4} a c e +\frac {1}{16} b^{2} e \right )}{4 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}+\frac {\left (a^{2} d -\frac {a \left (\frac {9}{7} a c d +\frac {3}{35} d \,b^{2}\right )}{3 c}\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 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (a^{2} e -\frac {a \left (\frac {5}{4} a c e +\frac {1}{16} b^{2} e \right )}{2 c}-\frac {b \left (\frac {5 a b e}{4}-\frac {3 b \left (\frac {5}{4} a c e +\frac {1}{16} b^{2} e \right )}{4 c}\right )}{2 c}\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 {\left (\frac {46 a b d}{35}-\frac {2 b \left (\frac {9}{7} a c d +\frac {3}{35} d \,b^{2}\right )}{3 c}\right ) 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 \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 )}\)	\(690\)
risch	\(\frac {\left (560 e \,x^{6} c^{3}+640 d \,x^{5} c^{3}+840 b e \,x^{4} c^{2}+1024 b d \,x^{3} c^{2}+1400 a \,c^{2} e \,x^{2}+70 b^{2} c e \,x^{2}+1920 a \,c^{2} d x +128 b^{2} c d x +700 a b c e -105 b^{3} e \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4480 c^{2}}+\frac {-\frac {128 c b d \left (8 a c -b^{2}\right ) 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 )}+\frac {105 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\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 {640 a^{2} c^{2} d \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 )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {32 a \,b^{2} c d \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 )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{4480 c^{2}}\)	\(695\)
default	\(d \left (\frac {c \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7}+\frac {8 b \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {\left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (a^{2}-\frac {\left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) a}{3 c}\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 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {46 a b}{35}-\frac {2 \left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) b}{3 c}\right ) 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 \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 )}\right )+e \left (\frac {b^{2} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c}+\frac {3 b^{4} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {5}{2}}}+\frac {5 b a \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 c}+\frac {c \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8}+\frac {3 b \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16}-\frac {3 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 c^{2}}+\frac {5 a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16}-\frac {3 b^{2} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {3 a^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 \sqrt {c}}\right )\)	\(717\)

Input:

int((e*x+d)*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/8*c*e*x^6*(c*x^4+b*x^2+a)^(1/2)+1/7*c*d*x^5*(c*x^4+b*x^2+a)^(1/2)+3/16*e 
*b*x^4*(c*x^4+b*x^2+a)^(1/2)+8/35*b*d*x^3*(c*x^4+b*x^2+a)^(1/2)+1/4*(5/4*a 
*c*e+1/16*b^2*e)/c*x^2*(c*x^4+b*x^2+a)^(1/2)+1/3*(9/7*a*c*d+3/35*d*b^2)/c* 
x*(c*x^4+b*x^2+a)^(1/2)+1/2*(5/4*a*b*e-3/4*b/c*(5/4*a*c*e+1/16*b^2*e))/c*( 
c*x^4+b*x^2+a)^(1/2)+1/4*(a^2*d-1/3*a/c*(9/7*a*c*d+3/35*d*b^2))*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*(a^2*e-1/2*a/c*(5/4*a*c*e+1/16*b^2*e)-1/2*b/c*(5/4 
*a*b*e-3/4*b/c*(5/4*a*c*e+1/16*b^2*e)))*ln((2*c*x^2+b)/c^(1/2)+2*(c*x^4+b* 
x^2+a)^(1/2))/c^(1/2)-1/2*(46/35*a*b*d-2/3*b/c*(9/7*a*c*d+3/35*d*b^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)))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.06 \[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {105 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} e x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) - 512 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c - 8 \, a b c^{2}\right )} d x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (b^{4} - 8 \, a b^{2} c\right )} d x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 256 \, \sqrt {\frac {1}{2}} {\left ({\left (2 \, b^{3} c + 20 \, a c^{3} - {\left (16 \, a b + b^{2}\right )} c^{2}\right )} d x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (2 \, b^{4} - 20 \, a b c^{2} - {\left (16 \, a b^{2} - b^{3}\right )} c\right )} d x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 4 \, {\left (560 \, c^{4} e x^{7} + 640 \, c^{4} d x^{6} + 840 \, b c^{3} e x^{5} + 1024 \, b c^{3} d x^{4} + 70 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} e x^{3} + 128 \, {\left (b^{2} c^{2} + 15 \, a c^{3}\right )} d x^{2} - 35 \, {\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} e x - 256 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} d\right )} \sqrt {c x^{4} + b x^{2} + a}}{17920 \, c^{3} x} \] Input:

integrate((e*x+d)*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

1/17920*(105*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c)*e*x*log(8*c^2*x^4 + 8* 
b*c*x^2 + b^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c) - 
 512*sqrt(1/2)*((b^3*c - 8*a*b*c^2)*d*x*sqrt((b^2 - 4*a*c)/c^2) - (b^4 - 8 
*a*b^2*c)*d*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e( 
arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt 
((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 256*sqrt(1/2)*((2*b^3*c + 20*a 
*c^3 - (16*a*b + b^2)*c^2)*d*x*sqrt((b^2 - 4*a*c)/c^2) - (2*b^4 - 20*a*b*c 
^2 - (16*a*b^2 - b^3)*c)*d*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b) 
/c)*elliptic_f(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x) 
, 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 4*(560*c^4*e*x^ 
7 + 640*c^4*d*x^6 + 840*b*c^3*e*x^5 + 1024*b*c^3*d*x^4 + 70*(b^2*c^2 + 20* 
a*c^3)*e*x^3 + 128*(b^2*c^2 + 15*a*c^3)*d*x^2 - 35*(3*b^3*c - 20*a*b*c^2)* 
e*x - 256*(b^3*c - 8*a*b*c^2)*d)*sqrt(c*x^4 + b*x^2 + a))/(c^3*x)
 

Sympy [F]

\[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (d + e x\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:

integrate((e*x+d)*(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral((d + e*x)*(a + b*x**2 + c*x**4)**(3/2), x)
 

Maxima [F]

\[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} \,d x } \] Input:

integrate((e*x+d)*(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d), x)
 

Giac [F]

\[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} \,d x } \] Input:

integrate((e*x+d)*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (d+e\,x\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:

int((d + e*x)*(a + b*x^2 + c*x^4)^(3/2),x)
 

Output:

int((d + e*x)*(a + b*x^2 + c*x^4)^(3/2), x)
 

Reduce [F]

\[ \int (d+e x) \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)*(c*x^4+b*x^2+a)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(1400*sqrt(a + b*x**2 + c*x**4)*a*b*c**2*e + 3840*sqrt(a + b*x**2 + c*x**4 
)*a*c**3*d*x + 2800*sqrt(a + b*x**2 + c*x**4)*a*c**3*e*x**2 - 210*sqrt(a + 
 b*x**2 + c*x**4)*b**3*c*e + 256*sqrt(a + b*x**2 + c*x**4)*b**2*c**2*d*x + 
 140*sqrt(a + b*x**2 + c*x**4)*b**2*c**2*e*x**2 + 2048*sqrt(a + b*x**2 + c 
*x**4)*b*c**3*d*x**3 + 1680*sqrt(a + b*x**2 + c*x**4)*b*c**3*e*x**4 + 1280 
*sqrt(a + b*x**2 + c*x**4)*c**4*d*x**5 + 1120*sqrt(a + b*x**2 + c*x**4)*c* 
*4*e*x**6 - 1680*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*a** 
2*c**2*e + 840*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*a*b** 
2*c*e - 105*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*b**4*e + 
 1680*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a**2*c**2*e - 
840*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a*b**2*c*e + 105 
*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*b**4*e + 5120*int(s 
qrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x 
**6),x)*a**3*c**3*d - 256*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 
 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a**2*b**2*c**2*d + 4096*int((sqrt(a 
 + b*x**2 + c*x**4)*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c* 
x**6),x)*a*b**2*c**3*d - 512*int((sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2 + 
2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*b**4*c**2*d + 9216*int((s 
qrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + 
 b*c*x**6),x)*a**2*b*c**3*d - 768*int((sqrt(a + b*x**2 + c*x**4)*x**2)/...