Optimal. Leaf size=214 \[ -\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 x^{5/2} (a+b x)}-\frac{2 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{3/2} (a+b x)}-\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{\sqrt{x} (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}+\frac{2 b^3 B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0800435, antiderivative size = 214, 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} \[ -\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 x^{5/2} (a+b x)}-\frac{2 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{3/2} (a+b x)}-\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{\sqrt{x} (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}+\frac{2 b^3 B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
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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^{9/2}} \, 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^{9/2}} \, 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^{9/2}}+\frac{a^2 b^3 (3 A b+a B)}{x^{7/2}}+\frac{3 a b^4 (A b+a B)}{x^{5/2}}+\frac{b^5 (A b+3 a B)}{x^{3/2}}+\frac{b^6 B}{\sqrt{x}}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 a^2 (3 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 a b (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{3/2} (a+b x)}-\frac{2 b^2 (A b+3 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b^3 B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0346976, size = 84, normalized size = 0.39 \[ -\frac{2 \sqrt{(a+b x)^2} \left (7 a^2 b x (3 A+5 B x)+a^3 (5 A+7 B x)+35 a b^2 x^2 (A+3 B x)+35 b^3 x^3 (A-B x)\right )}{35 x^{7/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 92, normalized size = 0.4 \begin{align*} -{\frac{-70\,B{x}^{4}{b}^{3}+70\,A{b}^{3}{x}^{3}+210\,B{x}^{3}a{b}^{2}+70\,A{x}^{2}a{b}^{2}+70\,B{x}^{2}{a}^{2}b+42\,A{a}^{2}bx+14\,{a}^{3}Bx+10\,A{a}^{3}}{35\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02415, size = 181, normalized size = 0.85 \begin{align*} \frac{2}{15} \, B{\left (\frac{15 \,{\left (b^{3} x^{2} - a b^{2} x\right )}}{x^{\frac{3}{2}}} - \frac{10 \,{\left (3 \, a b^{2} x^{2} + a^{2} b x\right )}}{x^{\frac{5}{2}}} - \frac{5 \, a^{2} b x^{2} + 3 \, a^{3} x}{x^{\frac{7}{2}}}\right )} - \frac{2}{105} \, A{\left (\frac{35 \,{\left (3 \, b^{3} x^{2} + a b^{2} x\right )}}{x^{\frac{5}{2}}} + \frac{14 \,{\left (5 \, a b^{2} x^{2} + 3 \, a^{2} b x\right )}}{x^{\frac{7}{2}}} + \frac{3 \,{\left (7 \, a^{2} b x^{2} + 5 \, a^{3} x\right )}}{x^{\frac{9}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36096, size = 166, normalized size = 0.78 \begin{align*} \frac{2 \,{\left (35 \, B b^{3} x^{4} - 5 \, A a^{3} - 35 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} - 35 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 7 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.14654, size = 167, normalized size = 0.78 \begin{align*} 2 \, B b^{3} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - \frac{2 \,{\left (105 \, B a b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 35 \, A b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 35 \, B a^{2} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 35 \, A a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 7 \, B a^{3} x \mathrm{sgn}\left (b x + a\right ) + 21 \, A a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 5 \, A a^{3} \mathrm{sgn}\left (b x + a\right )\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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