Optimal. Leaf size=220 \[ \frac{2 b^2 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{15 (a+b x)}+\frac{6 a b x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac{2 a^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{11 (a+b x)}+\frac{2 a^3 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 b^3 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0886076, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ \frac{2 b^2 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{15 (a+b x)}+\frac{6 a b x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac{2 a^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{11 (a+b x)}+\frac{2 a^3 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 b^3 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x^{7/2} \left (a b+b^2 x\right )^3 (A+B x) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a^3 A b^3 x^{7/2}+a^2 b^3 (3 A b+a B) x^{9/2}+3 a b^4 (A b+a B) x^{11/2}+b^5 (A b+3 a B) x^{13/2}+b^6 B x^{15/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{2 a^3 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 a^2 (3 A b+a B) x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{6 a b (A b+a B) x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac{2 b^2 (A b+3 a B) x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)}+\frac{2 b^3 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.040214, size = 89, normalized size = 0.4 \[ \frac{2 x^{9/2} \sqrt{(a+b x)^2} \left (2295 a^2 b x (13 A+11 B x)+1105 a^3 (11 A+9 B x)+1683 a b^2 x^2 (15 A+13 B x)+429 b^3 x^3 (17 A+15 B x)\right )}{109395 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 92, normalized size = 0.4 \begin{align*}{\frac{12870\,B{x}^{4}{b}^{3}+14586\,A{b}^{3}{x}^{3}+43758\,B{x}^{3}a{b}^{2}+50490\,A{x}^{2}a{b}^{2}+50490\,B{x}^{2}{a}^{2}b+59670\,A{a}^{2}bx+19890\,{a}^{3}Bx+24310\,A{a}^{3}}{109395\, \left ( bx+a \right ) ^{3}}{x}^{{\frac{9}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01148, size = 185, normalized size = 0.84 \begin{align*} \frac{2}{6435} \,{\left (33 \,{\left (13 \, b^{3} x^{2} + 15 \, a b^{2} x\right )} x^{\frac{11}{2}} + 90 \,{\left (11 \, a b^{2} x^{2} + 13 \, a^{2} b x\right )} x^{\frac{9}{2}} + 65 \,{\left (9 \, a^{2} b x^{2} + 11 \, a^{3} x\right )} x^{\frac{7}{2}}\right )} A + \frac{2}{36465} \,{\left (143 \,{\left (15 \, b^{3} x^{2} + 17 \, a b^{2} x\right )} x^{\frac{13}{2}} + 374 \,{\left (13 \, a b^{2} x^{2} + 15 \, a^{2} b x\right )} x^{\frac{11}{2}} + 255 \,{\left (11 \, a^{2} b x^{2} + 13 \, a^{3} x\right )} x^{\frac{9}{2}}\right )} B \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60648, size = 198, normalized size = 0.9 \begin{align*} \frac{2}{109395} \,{\left (6435 \, B b^{3} x^{8} + 12155 \, A a^{3} x^{4} + 7293 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 25245 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 9945 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12922, size = 169, normalized size = 0.77 \begin{align*} \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]