Optimal. Leaf size=118 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^{3/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0430058, antiderivative size = 118, 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} \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^{3/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{7/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right ) (A+B x)}{x^{7/2}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a A b}{x^{7/2}}+\frac{b (A b+a B)}{x^{5/2}}+\frac{b^2 B}{x^{3/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0255423, size = 48, normalized size = 0.41 \[ -\frac{2 \sqrt{(a+b x)^2} (a (3 A+5 B x)+5 b x (A+3 B x))}{15 x^{5/2} (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 44, normalized size = 0.4 \begin{align*} -{\frac{30\,Bb{x}^{2}+10\,Abx+10\,aBx+6\,aA}{15\,bx+15\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05705, size = 46, normalized size = 0.39 \begin{align*} -\frac{2 \,{\left (3 \, b x^{2} + a x\right )} B}{3 \, x^{\frac{5}{2}}} - \frac{2 \,{\left (5 \, b x^{2} + 3 \, a x\right )} A}{15 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.48838, size = 73, normalized size = 0.62 \begin{align*} -\frac{2 \,{\left (15 \, B b x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \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 = 1.14543, size = 69, normalized size = 0.58 \begin{align*} -\frac{2 \,{\left (15 \, B b x^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, B a x \mathrm{sgn}\left (b x + a\right ) + 5 \, A b x \mathrm{sgn}\left (b x + a\right ) + 3 \, A a \mathrm{sgn}\left (b x + a\right )\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]