Optimal. Leaf size=121 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (A b-2 a B)}{7 b^3}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 b^3}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0757351, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {770, 76} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (A b-2 a B)}{7 b^3}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 b^3}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 76
Rubi steps
\begin{align*} \int x (A+B x) \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 \left (a b+b^2 x\right )^5 (A+B x) \, 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 (-A b+a B) \left (a b+b^2 x\right )^5}{b^2}+\frac{(A b-2 a B) \left (a b+b^2 x\right )^6}{b^3}+\frac{B \left (a b+b^2 x\right )^7}{b^4}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a (A b-a B) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac{(A b-2 a B) (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac{B (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.0397976, size = 125, normalized size = 1.03 \[ \frac{x^2 \sqrt{(a+b x)^2} \left (84 a^3 b^2 x^2 (5 A+4 B x)+56 a^2 b^3 x^3 (6 A+5 B x)+70 a^4 b x (4 A+3 B x)+28 a^5 (3 A+2 B x)+20 a b^4 x^4 (7 A+6 B x)+3 b^5 x^5 (8 A+7 B x)\right )}{168 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 140, normalized size = 1.2 \begin{align*}{\frac{{x}^{2} \left ( 21\,B{b}^{5}{x}^{6}+24\,{x}^{5}A{b}^{5}+120\,{x}^{5}Ba{b}^{4}+140\,{x}^{4}Aa{b}^{4}+280\,{x}^{4}B{a}^{2}{b}^{3}+336\,A{a}^{2}{b}^{3}{x}^{3}+336\,B{a}^{3}{b}^{2}{x}^{3}+420\,{x}^{2}A{a}^{3}{b}^{2}+210\,{x}^{2}B{a}^{4}b+280\,xA{a}^{4}b+56\,xB{a}^{5}+84\,A{a}^{5} \right ) }{168\, \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 = 1.31314, size = 258, normalized size = 2.13 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} + \frac{1}{2} \, A a^{5} x^{2} + \frac{1}{7} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + \frac{5}{6} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac{5}{4} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \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 \left (A + B x\right ) \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 [B] time = 1.211, size = 298, normalized size = 2.46 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, B a b^{4} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, A b^{5} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, B a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, A a b^{4} x^{6} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a^{3} b^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{2} b^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, B a^{4} b x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, A a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B a^{5} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, A a^{4} b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a^{5} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (B a^{8} - 4 \, A a^{7} b\right )} \mathrm{sgn}\left (b x + a\right )}{168 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]