\(\int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx\) [128]

Optimal result

Integrand size = 19, antiderivative size = 629 \[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=-\frac {\sqrt {a+b (c+d x)^4}}{\left (a+b c^4\right ) x}+\frac {\sqrt {b} d (c+d x) \sqrt {a+b (c+d x)^4}}{\left (a+b c^4\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )}+\frac {b c^3 d \text {arctanh}\left (\frac {\sqrt {a+b c^4} (c+d x)}{c \sqrt {a+b (c+d x)^4}}\right )}{\left (a+b c^4\right )^{3/2}}+\frac {b c^3 d \text {arctanh}\left (\frac {a+b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\left (a+b c^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (a+b c^4\right ) \sqrt {a+b (c+d x)^4}}+\frac {\sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}}+\frac {b^{3/4} c^2 \left (\sqrt {a}-\sqrt {b} c^2\right ) d \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {a}+\sqrt {b} c^2\right )^2}{4 \sqrt {a} \sqrt {b} c^2},2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} c^2\right ) \left (a+b c^4\right ) \sqrt {a+b (c+d x)^4}} \] Output:

-(a+b*(d*x+c)^4)^(1/2)/(b*c^4+a)/x+b^(1/2)*d*(d*x+c)*(a+b*(d*x+c)^4)^(1/2) 
/(b*c^4+a)/(a^(1/2)+b^(1/2)*(d*x+c)^2)+b*c^3*d*arctanh((b*c^4+a)^(1/2)*(d* 
x+c)/c/(a+b*(d*x+c)^4)^(1/2))/(b*c^4+a)^(3/2)+b*c^3*d*arctanh((a+b*c^2*(d* 
x+c)^2)/(b*c^4+a)^(1/2)/(a+b*(d*x+c)^4)^(1/2))/(b*c^4+a)^(3/2)-a^(1/4)*b^( 
1/4)*d*(a^(1/2)+b^(1/2)*(d*x+c)^2)*((a+b*(d*x+c)^4)/(a^(1/2)+b^(1/2)*(d*x+ 
c)^2)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*(d*x+c)/a^(1/4))),1/2*2^(1/2 
))/(b*c^4+a)/(a+b*(d*x+c)^4)^(1/2)+1/2*b^(1/4)*d*(a^(1/2)+b^(1/2)*(d*x+c)^ 
2)*((a+b*(d*x+c)^4)/(a^(1/2)+b^(1/2)*(d*x+c)^2)^2)^(1/2)*InverseJacobiAM(2 
*arctan(b^(1/4)*(d*x+c)/a^(1/4)),1/2*2^(1/2))/a^(1/4)/(b^(1/2)*c^2+a^(1/2) 
)/(a+b*(d*x+c)^4)^(1/2)+1/2*b^(3/4)*c^2*(a^(1/2)-b^(1/2)*c^2)*d*(a^(1/2)+b 
^(1/2)*(d*x+c)^2)*((a+b*(d*x+c)^4)/(a^(1/2)+b^(1/2)*(d*x+c)^2)^2)^(1/2)*El 
lipticPi(sin(2*arctan(b^(1/4)*(d*x+c)/a^(1/4))),1/4*(b^(1/2)*c^2+a^(1/2))^ 
2/a^(1/2)/b^(1/2)/c^2,1/2*2^(1/2))/a^(1/4)/(b^(1/2)*c^2+a^(1/2))/(b*c^4+a) 
/(a+b*(d*x+c)^4)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.87 (sec) , antiderivative size = 1385, normalized size of antiderivative = 2.20 \[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx =\text {Too large to display} \] Input:

Integrate[1/(x^2*Sqrt[a + b*(c + d*x)^4]),x]
 

Output:

((-2*(a + b*(c + d*x)^4))/x + (4*(-1)^(1/4)*Sqrt[b]*c*d*((-1)^(1/4)*a^(1/4 
) - b^(1/4)*(c + d*x))^2*Sqrt[((1 - I)*((-1)^(3/4)*a^(1/4) - b^(1/4)*(c + 
d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*Sqrt[((-I)*((-1)^(1/4)*a^ 
(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*Sqrt 
[((-1 - I)*((-1)^(3/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - 
 b^(1/4)*(c + d*x))]*(((-1)^(1/4)*a^(1/4) - b^(1/4)*c)*EllipticF[ArcSin[Sq 
rt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b 
^(1/4)*(c + d*x))]], -1] - 2*(-1)^(1/4)*a^(1/4)*EllipticPi[-I, ArcSin[Sqrt 
[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^( 
1/4)*(c + d*x))]], -1]))/a^(1/4) + b^(1/4)*d*(2*((-1)^(3/4)*a^(3/4) + I*Sq 
rt[a]*b^(1/4)*(c + d*x) + (-1)^(1/4)*a^(1/4)*Sqrt[b]*(c + d*x)^2 + b^(3/4) 
*(c + d*x)^3) - ((2*I)*((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))^2*Sqrt[((1 
 - I)*((-1)^(3/4)*a^(1/4) - b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1 
/4)*(c + d*x))]*Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1) 
^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*Sqrt[((-1 - I)*((-1)^(3/4)*a^(1/4) + 
b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*((-1)^(3/4)* 
Sqrt[a]*EllipticE[ArcSin[Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x 
)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]], -1] + (-((-1)^(3/4)*Sqrt[a 
]) + 2*a^(1/4)*b^(1/4)*c + (-1)^(3/4)*Sqrt[b]*c^2)*EllipticF[ArcSin[Sqrt[( 
(-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^...
 

Rubi [A] (verified)

Time = 1.98 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.842, Rules used = {896, 2265, 25, 2280, 27, 1577, 488, 219, 2233, 25, 27, 1510, 2227, 27, 761, 2223}

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.

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

\(\Big \downarrow \) 896

\(\displaystyle d \int \frac {1}{d^2 x^2 \sqrt {b (c+d x)^4+a}}d(c+d x)\)

\(\Big \downarrow \) 2265

\(\displaystyle d \left (-\frac {b \int \frac {c^3+(c+d x) c^2+(c+d x)^2 c-(c+d x)^3}{d x \sqrt {b (c+d x)^4+a}}d(c+d x)}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle d \left (\frac {b \int -\frac {c^3+(c+d x) c^2+(c+d x)^2 c-(c+d x)^3}{d x \sqrt {b (c+d x)^4+a}}d(c+d x)}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 2280

\(\displaystyle d \left (\frac {b \left (\int \frac {c^4+2 (c+d x)^2 c^2-(c+d x)^4}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)+\int \frac {2 c^3 (c+d x)}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {b \left (\int \frac {c^4+2 (c+d x)^2 c^2-(c+d x)^4}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)+2 c^3 \int \frac {c+d x}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

\(\displaystyle d \left (\frac {b \left (\int \frac {c^4+2 (c+d x)^2 c^2-(c+d x)^4}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 2233

\(\displaystyle d \left (\frac {b \left (-\frac {\int -\frac {\sqrt {b} \left (c^2 \left (\sqrt {b} c^2+\sqrt {a}\right )-\left (\sqrt {a}-\sqrt {b} c^2\right ) (c+d x)^2\right )}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{b}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{\sqrt {a} \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle d \left (\frac {b \left (\frac {\int \frac {\sqrt {b} \left (c^2 \left (\sqrt {b} c^2+\sqrt {a}\right )-\left (\sqrt {a}-\sqrt {b} c^2\right ) (c+d x)^2\right )}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{b}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{\sqrt {a} \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {b \left (\frac {\int \frac {c^2 \left (\sqrt {b} c^2+\sqrt {a}\right )-\left (\sqrt {a}-\sqrt {b} c^2\right ) (c+d x)^2}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{\sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 1510

\(\displaystyle d \left (\frac {b \left (\frac {\int \frac {c^2 \left (\sqrt {b} c^2+\sqrt {a}\right )-\left (\sqrt {a}-\sqrt {b} c^2\right ) (c+d x)^2}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {b}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b (c+d x)^4}}-\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {a}+\sqrt {b} (c+d x)^2}}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 2227

\(\displaystyle d \left (\frac {b \left (\frac {\frac {2 \sqrt {a} \sqrt {b} c^4 \int \frac {\sqrt {b} (c+d x)^2+\sqrt {a}}{\sqrt {a} \left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {a}+\sqrt {b} c^2}+\frac {\left (a+b c^4\right ) \int \frac {1}{\sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {a}+\sqrt {b} c^2}}{\sqrt {b}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b (c+d x)^4}}-\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {a}+\sqrt {b} (c+d x)^2}}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {b \left (\frac {\frac {2 \sqrt {b} c^4 \int \frac {\sqrt {b} (c+d x)^2+\sqrt {a}}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {a}+\sqrt {b} c^2}+\frac {\left (a+b c^4\right ) \int \frac {1}{\sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {a}+\sqrt {b} c^2}}{\sqrt {b}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b (c+d x)^4}}-\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {a}+\sqrt {b} (c+d x)^2}}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle d \left (\frac {b \left (\frac {\frac {2 \sqrt {b} c^4 \int \frac {\sqrt {b} (c+d x)^2+\sqrt {a}}{\left (c^2-(c+d x)^2\right ) \sqrt {b (c+d x)^4+a}}d(c+d x)}{\sqrt {a}+\sqrt {b} c^2}+\frac {\left (a+b c^4\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}}}{\sqrt {b}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b (c+d x)^4}}-\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {a}+\sqrt {b} (c+d x)^2}}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

\(\Big \downarrow \) 2223

\(\displaystyle d \left (\frac {b \left (\frac {\frac {2 \sqrt {b} c^4 \left (\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b c^4} (c+d x)}{c \sqrt {a+b (c+d x)^4}}\right )}{2 c \sqrt {a+b c^4}}-\frac {\left (\sqrt {b}-\frac {\sqrt {a}}{c^2}\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} c^2+\sqrt {a}\right )^2}{4 \sqrt {a} \sqrt {b} c^2},2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a}+\sqrt {b} c^2}+\frac {\left (a+b c^4\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} c^2\right ) \sqrt {a+b (c+d x)^4}}}{\sqrt {b}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b (c+d x)^4}}-\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {a}+\sqrt {b} (c+d x)^2}}{\sqrt {b}}-\frac {c^3 \text {arctanh}\left (\frac {-a-b c^2 (c+d x)^2}{\sqrt {a+b c^4} \sqrt {a+b (c+d x)^4}}\right )}{\sqrt {a+b c^4}}\right )}{a+b c^4}-\frac {\sqrt {a+b (c+d x)^4}}{d x \left (a+b c^4\right )}\right )\)

Input:

Int[1/(x^2*Sqrt[a + b*(c + d*x)^4]),x]
 

Output:

d*(-(Sqrt[a + b*(c + d*x)^4]/((a + b*c^4)*d*x)) + (b*(-((c^3*ArcTanh[(-a - 
 b*c^2*(c + d*x)^2)/(Sqrt[a + b*c^4]*Sqrt[a + b*(c + d*x)^4])])/Sqrt[a + b 
*c^4]) - (-(((c + d*x)*Sqrt[a + b*(c + d*x)^4])/(Sqrt[a] + Sqrt[b]*(c + d* 
x)^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4) 
/(Sqrt[a] + Sqrt[b]*(c + d*x)^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*(c + d*x)) 
/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*(c + d*x)^4]))/Sqrt[b] + (((a + b*c^4 
)*(Sqrt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4)/(Sqrt[a] + Sqrt 
[b]*(c + d*x)^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*(c + d*x))/a^(1/4)], 1/2]) 
/(2*a^(1/4)*b^(1/4)*(Sqrt[a] + Sqrt[b]*c^2)*Sqrt[a + b*(c + d*x)^4]) + (2* 
Sqrt[b]*c^4*(((Sqrt[a] + Sqrt[b]*c^2)*ArcTanh[(Sqrt[a + b*c^4]*(c + d*x))/ 
(c*Sqrt[a + b*(c + d*x)^4])])/(2*c*Sqrt[a + b*c^4]) - ((Sqrt[b] - Sqrt[a]/ 
c^2)*(Sqrt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4)/(Sqrt[a] + S 
qrt[b]*(c + d*x)^2)^2]*EllipticPi[(Sqrt[a] + Sqrt[b]*c^2)^2/(4*Sqrt[a]*Sqr 
t[b]*c^2), 2*ArcTan[(b^(1/4)*(c + d*x))/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4) 
*Sqrt[a + b*(c + d*x)^4])))/(Sqrt[a] + Sqrt[b]*c^2))/Sqrt[b]))/(a + b*c^4) 
)
 

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 488

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

rule 761

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

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1510

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

rule 1577

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

rule 2223

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* 
(d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] 
)), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 
2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc 
Tan[q*x], 1/2], x]] /; FreeQ[{a, 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[c*(d/e) + a*(e 
/d)]
 

rule 2227

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

rule 2233

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff 
[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim 
p[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x 
^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 
2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2265

Int[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[e 
^3*(d + e*x)^(q + 1)*(Sqrt[a + c*x^4]/((q + 1)*(c*d^4 + a*e^4))), x] + Simp 
[c/((q + 1)*(c*d^4 + a*e^4))   Int[((d + e*x)^(q + 1)/Sqrt[a + c*x^4])*Simp 
[d^3*(q + 1) - d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x], x 
], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]
 

rule 2280

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Wit 
h[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff 
[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sqrt[a 
+ c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt 
[a + c*x^4]), x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px 
, x], 3] && NeQ[c*d^4 + a*e^4, 0]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.34 (sec) , antiderivative size = 5348, normalized size of antiderivative = 8.50

method	result	size
risch	\(\text {Expression too large to display}\)	\(5348\)
default	\(\text {Expression too large to display}\)	\(5365\)
elliptic	\(\text {Expression too large to display}\)	\(5365\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4}}}\, dx \] Input:

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

Output:

Integral(1/(x**2*sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b 
*c*d**3*x**3 + b*d**4*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=\int { \frac {1}{\sqrt {{\left (d x + c\right )}^{4} b + a} x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=\int { \frac {1}{\sqrt {{\left (d x + c\right )}^{4} b + a} x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=\int \frac {1}{x^2\,\sqrt {a+b\,{\left (c+d\,x\right )}^4}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt {a+b (c+d x)^4}} \, dx=\int \frac {1}{x^{2} \sqrt {a +b \left (d x +c \right )^{4}}}d x \] Input:

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

Output:

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