Optimal. Leaf size=336 \[ \frac{b x}{2 c \sqrt [4]{a+b x^2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \sqrt [4]{a+b x^2} (b c-a d)}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (3 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{d} x (a d-b c)^{3/2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (3 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{d} x (a d-b c)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.254961, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {414, 530, 229, 227, 196, 399, 490, 1218} \[ \frac{b x}{2 c \sqrt [4]{a+b x^2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \sqrt [4]{a+b x^2} (b c-a d)}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (3 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{d} x (a d-b c)^{3/2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (3 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{d} x (a d-b c)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 414
Rule 530
Rule 229
Rule 227
Rule 196
Rule 399
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx &=-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\int \frac{2 b c-a d+\frac{1}{2} b d x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{b \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{4 c (b c-a d)}+\frac{(3 b c-2 a d) \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\left ((3 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c (b c-a d) x}+\frac{\left (b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{4 c (b c-a d) \sqrt [4]{a+b x^2}}\\ &=\frac{b x}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{\left ((3 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}-\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c \sqrt{d} (b c-a d) x}+\frac{\left ((3 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}+\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c \sqrt{d} (b c-a d) x}-\frac{\left (b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{4 c (b c-a d) \sqrt [4]{a+b x^2}}\\ &=\frac{b x}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac{d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{\sqrt{a} \sqrt{b} \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac{\sqrt [4]{a} (3 b c-2 a d) \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{d} (-b c+a d)^{3/2} x}+\frac{\sqrt [4]{a} (3 b c-2 a d) \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{d} (-b c+a d)^{3/2} x}\\ \end{align*}
Mathematica [C] time = 0.242853, size = 392, normalized size = 1.17 \[ \frac{-d x^3 \left (6 c \left (a+b x^2\right )-b x^2 \sqrt [4]{\frac{b x^2}{a}+1} \left (c+d x^2\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right ) \left (b d x^2 \sqrt [4]{\frac{b x^2}{a}+1} \left (c+d x^2\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-6 c \left (2 a d-2 b c+b d x^2\right )\right )}{12 c^2 \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]