Optimal. Leaf size=85 \[ -\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0264572, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ -\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}-\frac{8 \int \frac{\left (a+b x^2\right )^{3/4}}{(c x)^{13/2}} \, dx}{3 a}\\ &=-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}+\frac{32 \int \frac{\left (a+b x^2\right )^{7/4}}{(c x)^{13/2}} \, dx}{21 a^2}\\ &=-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0237282, size = 52, normalized size = 0.61 \[ -\frac{2 \sqrt{c x} \left (a+b x^2\right )^{3/4} \left (21 a^2-24 a b x^2+32 b^2 x^4\right )}{231 a^3 c^7 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 42, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 32\,{b}^{2}{x}^{4}-24\,ab{x}^{2}+21\,{a}^{2} \right ) }{231\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{4}}} \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{1}{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 = 1.44771, size = 115, normalized size = 1.35 \begin{align*} -\frac{2 \,{\left (32 \, b^{2} x^{4} - 24 \, a b x^{2} + 21 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{231 \, a^{3} c^{7} x^{6}} \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{1}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]