Optimal. Leaf size=236 \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{e^4 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{3 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) \log (d+e x)}{e^5 (a+b x)}+\frac{B (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b e} \]
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Rubi [A] time = 0.168176, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{e^4 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{3 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) \log (d+e x)}{e^5 (a+b x)}+\frac{B (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b e} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, 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)}{d+e x} \, 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{b^4 (b d-a e)^2 (-B d+A e)}{e^4}-\frac{b^4 (b d-a e) (-B d+A e) (a+b x)}{e^3}+\frac{b^4 (-B d+A e) (a+b x)^2}{e^2}+\frac{B \left (a b+b^2 x\right )^3}{e}-\frac{b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{b (b d-a e)^2 (B d-A e) x \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac{(b d-a e) (B d-A e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3}-\frac{(B d-A e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2}+\frac{B (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b e}+\frac{(b d-a e)^3 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.126725, size = 187, normalized size = 0.79 \[ \frac{\sqrt{(a+b x)^2} \left (e x \left (18 a^2 b e^2 (2 A e-2 B d+B e x)+12 a^3 B e^3+6 a b^2 e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )\right )+12 (b d-a e)^3 (B d-A e) \log (d+e x)\right )}{12 e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 358, normalized size = 1.5 \begin{align*}{\frac{3\,B{x}^{4}{b}^{3}{e}^{4}+4\,A{x}^{3}{b}^{3}{e}^{4}+12\,B{x}^{3}a{b}^{2}{e}^{4}-4\,B{x}^{3}{b}^{3}d{e}^{3}+18\,A{x}^{2}a{b}^{2}{e}^{4}-6\,A{x}^{2}{b}^{3}d{e}^{3}+18\,B{x}^{2}{a}^{2}b{e}^{4}-18\,B{x}^{2}a{b}^{2}d{e}^{3}+6\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+12\,A\ln \left ( ex+d \right ){a}^{3}{e}^{4}-36\,A\ln \left ( ex+d \right ){a}^{2}bd{e}^{3}+36\,A\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}{e}^{2}-12\,A\ln \left ( ex+d \right ){b}^{3}{d}^{3}e+36\,Ax{a}^{2}b{e}^{4}-36\,Axa{b}^{2}d{e}^{3}+12\,Ax{b}^{3}{d}^{2}{e}^{2}-12\,B\ln \left ( ex+d \right ){a}^{3}d{e}^{3}+36\,B\ln \left ( ex+d \right ){a}^{2}b{d}^{2}{e}^{2}-36\,B\ln \left ( ex+d \right ) a{b}^{2}{d}^{3}e+12\,B\ln \left ( ex+d \right ){b}^{3}{d}^{4}+12\,Bx{a}^{3}{e}^{4}-36\,Bx{a}^{2}bd{e}^{3}+36\,Bxa{b}^{2}{d}^{2}{e}^{2}-12\,Bx{b}^{3}{d}^{3}e}{12\, \left ( bx+a \right ) ^{3}{e}^{5}} \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 = 1.5632, size = 531, normalized size = 2.25 \begin{align*} \frac{3 \, B b^{3} e^{4} x^{4} - 4 \,{\left (B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 6 \,{\left (B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 12 \,{\left (B b^{3} d^{3} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 12 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \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}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18073, size = 578, normalized size = 2.45 \begin{align*}{\left (B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) - 3 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) - A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + A a^{3} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, B b^{3} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) - 4 \, B b^{3} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, B b^{3} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) - 12 \, B b^{3} d^{3} x \mathrm{sgn}\left (b x + a\right ) + 12 \, B a b^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 4 \, A b^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 18 \, B a b^{2} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 6 \, A b^{3} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 36 \, B a b^{2} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + 12 \, A b^{3} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + 18 \, B a^{2} b x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 18 \, A a b^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 36 \, B a^{2} b d x e^{2} \mathrm{sgn}\left (b x + a\right ) - 36 \, A a b^{2} d x e^{2} \mathrm{sgn}\left (b x + a\right ) + 12 \, B a^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 36 \, A a^{2} b x e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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