Optimal. Leaf size=130 \[ -\frac{8 b^{5/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{7 a^{5/2} c^6 \left (a-b x^2\right )^{3/4}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0943923, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {325, 329, 237, 335, 275, 232} \[ -\frac{8 b^{5/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a-b x^2\right )^{3/4}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 329
Rule 237
Rule 335
Rule 275
Rule 232
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{9/2} \left (a-b x^2\right )^{3/4}} \, dx &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}+\frac{(6 b) \int \frac{1}{(c x)^{5/2} \left (a-b x^2\right )^{3/4}} \, dx}{7 a c^2}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac{\left (4 b^2\right ) \int \frac{1}{\sqrt{c x} \left (a-b x^2\right )^{3/4}} \, dx}{7 a^2 c^4}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a-\frac{b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt{c x}\right )}{7 a^2 c^5}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac{\left (8 b^2 \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt{c x}\right )}{7 a^2 c^5 \left (a-b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{\left (8 b^2 \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{\sqrt{c x}}\right )}{7 a^2 c^5 \left (a-b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{\left (4 b^2 \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{c x}\right )}{7 a^2 c^5 \left (a-b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac{4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{8 b^{5/2} \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a-b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0126375, size = 57, normalized size = 0.44 \[ -\frac{2 x \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (-\frac{7}{4},\frac{3}{4};-\frac{3}{4};\frac{b x^2}{a}\right )}{7 (c x)^{9/2} \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{9}{2}}} \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{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{9}{2}}}\,{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{1}{4}} \sqrt{c x}}{b c^{5} x^{7} - a c^{5} x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]