Optimal. Leaf size=128 \[ \frac{3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac{3 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0450154, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {277, 325, 229, 228} \[ \frac{3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac{3 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 277
Rule 325
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{\left (a-b x^2\right )^{3/4}}{x^6} \, dx &=-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5}-\frac{1}{10} (3 b) \int \frac{1}{x^4 \sqrt [4]{a-b x^2}} \, dx\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac{b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac{\left (3 b^2\right ) \int \frac{1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac{b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac{3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac{\left (3 b^3\right ) \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^2}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac{b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac{3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac{\left (3 b^3 \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{40 a^2 \sqrt [4]{a-b x^2}}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac{b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac{3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac{3 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0094402, size = 52, normalized size = 0.41 \[ -\frac{\left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac{5}{2},-\frac{3}{4};-\frac{3}{2};\frac{b x^2}{a}\right )}{5 x^5 \left (1-\frac{b x^2}{a}\right )^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{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{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.45972, size = 36, normalized size = 0.28 \begin{align*} - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]