Optimal. Leaf size=114 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0458225, antiderivative size = 114, 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 = {770, 76} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
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^6} \, 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^6} \, 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^6}+\frac{b (A b+a B)}{x^5}+\frac{b^2 B}{x^4}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{(A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.015919, size = 49, normalized size = 0.43 \[ -\frac{\sqrt{(a+b x)^2} (3 a (4 A+5 B x)+5 b x (3 A+4 B x))}{60 x^5 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 44, normalized size = 0.4 \begin{align*} -{\frac{20\,Bb{x}^{2}+15\,Abx+15\,aBx+12\,aA}{60\,{x}^{5} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{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 = 1.57358, size = 70, normalized size = 0.61 \begin{align*} -\frac{20 \, B b x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.913517, size = 31, normalized size = 0.27 \begin{align*} - \frac{12 A a + 20 B b x^{2} + x \left (15 A b + 15 B a\right )}{60 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15266, size = 104, normalized size = 0.91 \begin{align*} -\frac{{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} \mathrm{sgn}\left (b x + a\right )}{60 \, a^{4}} - \frac{20 \, B b x^{2} \mathrm{sgn}\left (b x + a\right ) + 15 \, B a x \mathrm{sgn}\left (b x + a\right ) + 15 \, A b x \mathrm{sgn}\left (b x + a\right ) + 12 \, A a \mathrm{sgn}\left (b x + a\right )}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]