Optimal. Leaf size=231 \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0528161, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 43} \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{11}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{x^{11}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^5 b^5}{x^{11}}+\frac{5 a^4 b^6}{x^{10}}+\frac{10 a^3 b^7}{x^9}+\frac{10 a^2 b^8}{x^8}+\frac{5 a b^9}{x^7}+\frac{b^{10}}{x^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0159684, size = 77, normalized size = 0.33 \[ -\frac{\sqrt{(a+b x)^2} \left (1575 a^3 b^2 x^2+1800 a^2 b^3 x^3+700 a^4 b x+126 a^5+1050 a b^4 x^4+252 b^5 x^5\right )}{1260 x^{10} (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.183, size = 74, normalized size = 0.3 \begin{align*} -{\frac{252\,{b}^{5}{x}^{5}+1050\,a{b}^{4}{x}^{4}+1800\,{a}^{2}{b}^{3}{x}^{3}+1575\,{a}^{3}{b}^{2}{x}^{2}+700\,{a}^{4}bx+126\,{a}^{5}}{1260\,{x}^{10} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \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 = 1.64799, size = 144, normalized size = 0.62 \begin{align*} -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \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 (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{11}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30161, size = 146, normalized size = 0.63 \begin{align*} -\frac{b^{10} \mathrm{sgn}\left (b x + a\right )}{1260 \, a^{5}} - \frac{252 \, b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + 1050 \, a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 1800 \, a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 1575 \, a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 700 \, a^{4} b x \mathrm{sgn}\left (b x + a\right ) + 126 \, a^{5} \mathrm{sgn}\left (b x + a\right )}{1260 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]