Optimal. Leaf size=109 \[ -\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0374668, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ -\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx &=\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}}+\frac{12 \int \frac{1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx}{a}\\ &=\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}-\frac{32 \int \frac{\left (a+b x^2\right )^{3/4}}{(c x)^{13/2}} \, dx}{a^2}\\ &=\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}+\frac{128 \int \frac{\left (a+b x^2\right )^{7/4}}{(c x)^{13/2}} \, dx}{7 a^3}\\ &=\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0122965, size = 58, normalized size = 0.53 \[ -\frac{2 x \left (-12 a^2 b x^2+7 a^3+32 a b^2 x^4+128 b^3 x^6\right )}{77 a^4 (c x)^{13/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 53, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 128\,{b}^{3}{x}^{6}+32\,a{b}^{2}{x}^{4}-12\,{a}^{2}b{x}^{2}+7\,{a}^{3} \right ) }{77\,{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( cx \right ) ^{-{\frac{13}{2}}}} \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 (c x\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61597, size = 158, normalized size = 1.45 \begin{align*} -\frac{2 \,{\left (128 \, b^{3} x^{6} + 32 \, a b^{2} x^{4} - 12 \, a^{2} b x^{2} + 7 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{77 \,{\left (a^{4} b c^{7} x^{8} + a^{5} c^{7} x^{6}\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{5}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]