Optimal. Leaf size=74 \[ -\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac{8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac{2 \left (a x+b x^2\right )^{7/2}}{11 a x^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0281318, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ -\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac{8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac{2 \left (a x+b x^2\right )^{7/2}}{11 a x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^9} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}-\frac{(4 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{11 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac{8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}+\frac{\left (8 b^2\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{99 a^2}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac{8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}\\ \end{align*}
Mathematica [A] time = 0.0126369, size = 47, normalized size = 0.64 \[ -\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (63 a^2-28 a b x+8 b^2 x^2\right )}{693 a^3 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 8\,{b}^{2}{x}^{2}-28\,abx+63\,{a}^{2} \right ) }{693\,{x}^{8}{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.04175, size = 161, normalized size = 2.18 \begin{align*} -\frac{2 \,{\left (8 \, b^{5} x^{5} - 4 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + 113 \, a^{3} b^{2} x^{2} + 161 \, a^{4} b x + 63 \, a^{5}\right )} \sqrt{b x^{2} + a x}}{693 \, a^{3} x^{6}} \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{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.36916, size = 340, normalized size = 4.59 \begin{align*} \frac{2 \,{\left (924 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} b^{4} + 4851 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a b^{\frac{7}{2}} + 11781 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{2} b^{3} + 16863 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{3} b^{\frac{5}{2}} + 15345 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{4} b^{2} + 9009 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{5} b^{\frac{3}{2}} + 3311 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{6} b + 693 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{7} \sqrt{b} + 63 \, a^{8}\right )}}{693 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]