Optimal. Leaf size=107 \[ -\frac{2 a^3 (a B+4 A b)}{\sqrt{x}}+4 a^2 b \sqrt{x} (2 a B+3 A b)-\frac{2 a^4 A}{3 x^{3/2}}+\frac{4}{3} a b^2 x^{3/2} (3 a B+2 A b)+\frac{2}{5} b^3 x^{5/2} (4 a B+A b)+\frac{2}{7} b^4 B x^{7/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0502736, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 76} \[ -\frac{2 a^3 (a B+4 A b)}{\sqrt{x}}+4 a^2 b \sqrt{x} (2 a B+3 A b)-\frac{2 a^4 A}{3 x^{3/2}}+\frac{4}{3} a b^2 x^{3/2} (3 a B+2 A b)+\frac{2}{5} b^3 x^{5/2} (4 a B+A b)+\frac{2}{7} b^4 B x^{7/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{5/2}} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{x^{5/2}} \, dx\\ &=\int \left (\frac{a^4 A}{x^{5/2}}+\frac{a^3 (4 A b+a B)}{x^{3/2}}+\frac{2 a^2 b (3 A b+2 a B)}{\sqrt{x}}+2 a b^2 (2 A b+3 a B) \sqrt{x}+b^3 (A b+4 a B) x^{3/2}+b^4 B x^{5/2}\right ) \, dx\\ &=-\frac{2 a^4 A}{3 x^{3/2}}-\frac{2 a^3 (4 A b+a B)}{\sqrt{x}}+4 a^2 b (3 A b+2 a B) \sqrt{x}+\frac{4}{3} a b^2 (2 A b+3 a B) x^{3/2}+\frac{2}{5} b^3 (A b+4 a B) x^{5/2}+\frac{2}{7} b^4 B x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0305323, size = 86, normalized size = 0.8 \[ \frac{420 a^2 b^2 x^2 (3 A+B x)+840 a^3 b x (B x-A)-70 a^4 (A+3 B x)+56 a b^3 x^3 (5 A+3 B x)+6 b^4 x^4 (7 A+5 B x)}{105 x^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 100, normalized size = 0.9 \begin{align*} -{\frac{-30\,{b}^{4}B{x}^{5}-42\,A{b}^{4}{x}^{4}-168\,B{x}^{4}a{b}^{3}-280\,aA{b}^{3}{x}^{3}-420\,B{x}^{3}{a}^{2}{b}^{2}-1260\,{a}^{2}A{b}^{2}{x}^{2}-840\,B{x}^{2}{a}^{3}b+840\,{a}^{3}Abx+210\,{a}^{4}Bx+70\,A{a}^{4}}{105}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01173, size = 134, normalized size = 1.25 \begin{align*} \frac{2}{7} \, B b^{4} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{5}{2}} + \frac{4}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{3}{2}} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{4} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57936, size = 228, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (15 \, B b^{4} x^{5} - 35 \, A a^{4} + 21 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 70 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 105 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{105 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.54897, size = 139, normalized size = 1.3 \begin{align*} - \frac{2 A a^{4}}{3 x^{\frac{3}{2}}} - \frac{8 A a^{3} b}{\sqrt{x}} + 12 A a^{2} b^{2} \sqrt{x} + \frac{8 A a b^{3} x^{\frac{3}{2}}}{3} + \frac{2 A b^{4} x^{\frac{5}{2}}}{5} - \frac{2 B a^{4}}{\sqrt{x}} + 8 B a^{3} b \sqrt{x} + 4 B a^{2} b^{2} x^{\frac{3}{2}} + \frac{8 B a b^{3} x^{\frac{5}{2}}}{5} + \frac{2 B b^{4} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21814, size = 134, normalized size = 1.25 \begin{align*} \frac{2}{7} \, B b^{4} x^{\frac{7}{2}} + \frac{8}{5} \, B a b^{3} x^{\frac{5}{2}} + \frac{2}{5} \, A b^{4} x^{\frac{5}{2}} + 4 \, B a^{2} b^{2} x^{\frac{3}{2}} + \frac{8}{3} \, A a b^{3} x^{\frac{3}{2}} + 8 \, B a^{3} b \sqrt{x} + 12 \, A a^{2} b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{4} x + 12 \, A a^{3} b x + A a^{4}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]