Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{5} b x^{5/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{7} b^2 B x^{7/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0273857, antiderivative size = 61, 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 = {27, 76} \[ 2 a^2 A \sqrt{x}+\frac{2}{5} b x^{5/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{7} b^2 B x^{7/2} \]
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 )}{\sqrt{x}} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{\sqrt{x}} \, dx\\ &=\int \left (\frac{a^2 A}{\sqrt{x}}+a (2 A b+a B) \sqrt{x}+b (A b+2 a B) x^{3/2}+b^2 B x^{5/2}\right ) \, dx\\ &=2 a^2 A \sqrt{x}+\frac{2}{3} a (2 A b+a B) x^{3/2}+\frac{2}{5} b (A b+2 a B) x^{5/2}+\frac{2}{7} b^2 B x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0167669, size = 51, normalized size = 0.84 \[ \frac{2}{105} \sqrt{x} \left (35 a^2 (3 A+B x)+14 a b x (5 A+3 B x)+3 b^2 x^2 (7 A+5 B x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 52, normalized size = 0.9 \begin{align*}{\frac{30\,{b}^{2}B{x}^{3}+42\,A{b}^{2}{x}^{2}+84\,B{x}^{2}ab+140\,aAbx+70\,{a}^{2}Bx+210\,A{a}^{2}}{105}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03928, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.48841, size = 127, normalized size = 2.08 \begin{align*} \frac{2}{105} \,{\left (15 \, B b^{2} x^{3} + 105 \, A a^{2} + 21 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.531826, size = 78, normalized size = 1.28 \begin{align*} 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31061, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B a b x^{\frac{5}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} + 2 \, A a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]