Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{8 x^8 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 x^7 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 x^6 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
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Rubi [A] time = 0.0777503, 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} \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{8 x^8 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 x^7 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 x^6 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
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^{10}} \, 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^{10}} \, 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^{10}}+\frac{a^2 b^3 (3 A b+a B)}{x^9}+\frac{3 a b^4 (A b+a B)}{x^8}+\frac{b^5 (A b+3 a B)}{x^7}+\frac{b^6 B}{x^6}\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}}{9 x^9 (a+b x)}-\frac{a^2 (3 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac{3 a b (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{b^2 (A b+3 a B) \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}}{5 x^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0288099, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (135 a^2 b x (7 A+8 B x)+35 a^3 (8 A+9 B x)+180 a b^2 x^2 (6 A+7 B x)+84 b^3 x^3 (5 A+6 B x)\right )}{2520 x^9 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 92, normalized size = 0.4 \begin{align*} -{\frac{504\,B{x}^{4}{b}^{3}+420\,A{b}^{3}{x}^{3}+1260\,B{x}^{3}a{b}^{2}+1080\,A{x}^{2}a{b}^{2}+1080\,B{x}^{2}{a}^{2}b+945\,A{a}^{2}bx+315\,{a}^{3}Bx+280\,A{a}^{3}}{2520\,{x}^{9} \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.
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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.
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Fricas [A] time = 1.31769, size = 176, normalized size = 0.84 \begin{align*} -\frac{504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21294, size = 201, normalized size = 0.96 \begin{align*} -\frac{{\left (9 \, B a b^{8} - 5 \, A b^{9}\right )} \mathrm{sgn}\left (b x + a\right )}{2520 \, a^{6}} - \frac{504 \, B b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 1260 \, B a b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 420 \, A b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 1080 \, B a^{2} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 1080 \, A a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 315 \, B a^{3} x \mathrm{sgn}\left (b x + a\right ) + 945 \, A a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 280 \, A a^{3} \mathrm{sgn}\left (b x + a\right )}{2520 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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