Optimal. Leaf size=231 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^5 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.061848, 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} \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^5 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x^5 \left (a b+b^2 x\right )^5 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a^5 b^5 x^5+5 a^4 b^6 x^6+10 a^3 b^7 x^7+10 a^2 b^8 x^8+5 a b^9 x^9+b^{10} x^{10}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{a^5 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^5 x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0229274, size = 77, normalized size = 0.33 \[ \frac{x^6 \sqrt{(a+b x)^2} \left (3465 a^3 b^2 x^2+3080 a^2 b^3 x^3+1980 a^4 b x+462 a^5+1386 a b^4 x^4+252 b^5 x^5\right )}{2772 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.176, size = 74, normalized size = 0.3 \begin{align*}{\frac{{x}^{6} \left ( 252\,{b}^{5}{x}^{5}+1386\,a{b}^{4}{x}^{4}+3080\,{a}^{2}{b}^{3}{x}^{3}+3465\,{a}^{3}{b}^{2}{x}^{2}+1980\,{a}^{4}bx+462\,{a}^{5} \right ) }{2772\, \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 = 1.55523, size = 134, normalized size = 0.58 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{6} \, a^{5} 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 x^{5} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22666, size = 144, normalized size = 0.62 \begin{align*} \frac{1}{11} \, b^{5} x^{11} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a b^{4} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{9} \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, a^{3} b^{2} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, a^{4} b x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, a^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) - \frac{a^{11} \mathrm{sgn}\left (b x + a\right )}{2772 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]