Optimal. Leaf size=144 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}-\frac{3 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0519569, antiderivative size = 144, 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} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}-\frac{3 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^3 \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^3 \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 (-\frac{a^3 \left (a b+b^2 x\right )^5}{b^3}+\frac{3 a^2 \left (a b+b^2 x\right )^6}{b^4}-\frac{3 a \left (a b+b^2 x\right )^7}{b^5}+\frac{\left (a b+b^2 x\right )^8}{b^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a^3 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac{3 a^2 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^4}-\frac{3 a (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac{(a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^4}\\ \end{align*}
Mathematica [A] time = 0.0190502, size = 77, normalized size = 0.53 \[ \frac{x^4 \sqrt{(a+b x)^2} \left (840 a^3 b^2 x^2+720 a^2 b^3 x^3+504 a^4 b x+126 a^5+315 a b^4 x^4+56 b^5 x^5\right )}{504 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.188, size = 74, normalized size = 0.5 \begin{align*}{\frac{{x}^{4} \left ( 56\,{b}^{5}{x}^{5}+315\,a{b}^{4}{x}^{4}+720\,{a}^{2}{b}^{3}{x}^{3}+840\,{a}^{3}{b}^{2}{x}^{2}+504\,{a}^{4}bx+126\,{a}^{5} \right ) }{504\, \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 = 1.7779, size = 124, normalized size = 0.86 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{8} \, a b^{4} x^{8} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac{1}{4} \, a^{5} x^{4} \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^{3} \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 = 1.34331, size = 143, normalized size = 0.99 \begin{align*} \frac{1}{9} \, b^{5} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{8} \, a b^{4} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, a^{2} b^{3} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + a^{4} b x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a^{5} x^{4} \mathrm{sgn}\left (b x + a\right ) - \frac{a^{9} \mathrm{sgn}\left (b x + a\right )}{504 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]