Optimal. Leaf size=233 \[ \frac{2 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a+b x^2} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{3/2}}-\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.160015, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {403, 197, 196, 399, 490, 1218} \[ \frac{2 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a+b x^2} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{3/2}}-\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 403
Rule 197
Rule 196
Rule 399
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx &=\frac{b \int \frac{1}{\left (a+b x^2\right )^{5/4}} \, dx}{b c-a d}-\frac{d \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{b c-a d}\\ &=-\frac{\left (2 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 )}{(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}{a (b c-a d) \sqrt [4]{a+b x^2}}\\ &=\frac{2 \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 )}{\sqrt{a} (b c-a d) \sqrt [4]{a+b x^2}}+\frac{\left (\sqrt{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 )}{(b c-a d) x}-\frac{\left (\sqrt{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 )}{(b c-a d) x}\\ &=\frac{2 \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 )}{\sqrt{a} (b c-a d) \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{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 )}{(-b c+a d)^{3/2} x}-\frac{\sqrt [4]{a} \sqrt{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 )}{(-b c+a d)^{3/2} x}\\ \end{align*}
Mathematica [C] time = 0.268941, size = 327, normalized size = 1.4 \[ \frac{x \left (\frac{b d x^2 \sqrt [4]{\frac{b x^2}{a}+1} F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c}+\frac{6 \left (b x^2 \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+3 a c \left (a d-b \left (c+2 d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{\left (c+d x^2\right ) \left (6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}\right )}{3 a \sqrt [4]{a+b x^2} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, 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{5}{4}}{\left (d x^{2} + c\right )}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}} \left (c + d x^{2}\right )}\, dx \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{5}{4}}{\left (d x^{2} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]