Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 x^5 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 x^3 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0789072, antiderivative size = 210, 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} \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 x^5 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 x^3 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^7} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3 (A+B x)}{x^7} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^3 A b^3}{x^7}+\frac{a^2 b^3 (3 A b+a B)}{x^6}+\frac{3 a b^4 (A b+a B)}{x^5}+\frac{b^5 (A b+3 a B)}{x^4}+\frac{b^6 B}{x^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{a^2 (3 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{3 a b (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b^2 (A b+3 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0279849, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (9 a^2 b x (4 A+5 B x)+2 a^3 (5 A+6 B x)+15 a b^2 x^2 (3 A+4 B x)+10 b^3 x^3 (2 A+3 B x)\right )}{60 x^6 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 92, normalized size = 0.4 \begin{align*} -{\frac{30\,B{x}^{4}{b}^{3}+20\,A{b}^{3}{x}^{3}+60\,B{x}^{3}a{b}^{2}+45\,A{x}^{2}a{b}^{2}+45\,B{x}^{2}{a}^{2}b+36\,A{a}^{2}bx+12\,{a}^{3}Bx+10\,A{a}^{3}}{60\,{x}^{6} \left ( bx+a \right ) ^{3}} \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 [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 = 1.32112, size = 165, normalized size = 0.79 \begin{align*} -\frac{30 \, B b^{3} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.32415, size = 201, normalized size = 0.96 \begin{align*} \frac{{\left (3 \, B a b^{5} - A b^{6}\right )} \mathrm{sgn}\left (b x + a\right )}{60 \, a^{3}} - \frac{30 \, B b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 60 \, B a b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 20 \, A b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 45 \, B a^{2} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 45 \, A a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 12 \, B a^{3} x \mathrm{sgn}\left (b x + a\right ) + 36 \, A a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 10 \, A a^{3} \mathrm{sgn}\left (b x + a\right )}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]