Optimal. Leaf size=92 \[ -\frac{2 (d+e x)^2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{8 (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{3 b^4 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0518084, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {804, 636} \[ -\frac{2 (d+e x)^2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{8 (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{3 b^4 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 804
Rule 636
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (A b-(b B-2 A c) x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{(4 (b B d-2 A c d+A b e)) \int \frac{d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac{2 (A b-(b B-2 A c) x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{8 (b B d-2 A c d+A b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0854175, size = 149, normalized size = 1.62 \[ \frac{2 \left (A \left (2 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )+b^3 \left (-\left (d^2+6 d e x-3 e^2 x^2\right )\right )+8 b c^2 d x^2 (3 d-2 e x)+16 c^3 d^2 x^3\right )+b B x \left (b^2 \left (-3 d^2+6 d e x+e^2 x^2\right )+4 b c d x (e x-3 d)-8 c^2 d^2 x^2\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 197, normalized size = 2.1 \begin{align*} -{\frac{2\,x \left ( cx+b \right ) \left ( -2\,A{b}^{2}c{e}^{2}{x}^{3}+16\,Ab{c}^{2}de{x}^{3}-16\,A{c}^{3}{d}^{2}{x}^{3}-B{b}^{3}{e}^{2}{x}^{3}-4\,B{b}^{2}cde{x}^{3}+8\,Bb{c}^{2}{d}^{2}{x}^{3}-3\,A{b}^{3}{e}^{2}{x}^{2}+24\,A{b}^{2}cde{x}^{2}-24\,Ab{c}^{2}{d}^{2}{x}^{2}-6\,B{b}^{3}de{x}^{2}+12\,B{b}^{2}c{d}^{2}{x}^{2}+6\,A{b}^{3}dex-6\,A{b}^{2}c{d}^{2}x+3\,B{b}^{3}{d}^{2}x+A{d}^{2}{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0911, size = 468, normalized size = 5.09 \begin{align*} -\frac{B e^{2} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{4 \, A c d^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, A c^{2} d^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} - \frac{B b e^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c^{2}} + \frac{2 \, B e^{2} x}{3 \, \sqrt{c x^{2} + b x} b c} - \frac{2 \, A d^{2}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, A c d^{2}}{3 \, \sqrt{c x^{2} + b x} b^{3}} + \frac{B e^{2}}{3 \, \sqrt{c x^{2} + b x} c^{2}} + \frac{4 \,{\left (2 \, B d e + A e^{2}\right )} x}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \,{\left (B d^{2} + 2 \, A d e\right )} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{2 \,{\left (2 \, B d e + A e^{2}\right )} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{16 \,{\left (B d^{2} + 2 \, A d e\right )} c x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \,{\left (B d^{2} + 2 \, A d e\right )}}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \,{\left (2 \, B d e + A e^{2}\right )}}{3 \, \sqrt{c x^{2} + b x} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84641, size = 389, normalized size = 4.23 \begin{align*} -\frac{2 \,{\left (A b^{3} d^{2} +{\left (8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \,{\left (B b^{2} c - 4 \, A b c^{2}\right )} d e -{\left (B b^{3} + 2 \, A b^{2} c\right )} e^{2}\right )} x^{3} - 3 \,{\left (A b^{3} e^{2} - 4 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} + 2 \,{\left (B b^{3} - 4 \, A b^{2} c\right )} d e\right )} x^{2} + 3 \,{\left (2 \, A b^{3} d e +{\left (B b^{3} - 2 \, A b^{2} c\right )} d^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \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 (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29368, size = 254, normalized size = 2.76 \begin{align*} -\frac{{\left (x{\left (\frac{{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 16 \, A b c^{2} d e - B b^{3} e^{2} - 2 \, A b^{2} c e^{2}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (4 \, B b^{2} c d^{2} - 8 \, A b c^{2} d^{2} - 2 \, B b^{3} d e + 8 \, A b^{2} c d e - A b^{3} e^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (B b^{3} d^{2} - 2 \, A b^{2} c d^{2} + 2 \, A b^{3} d e\right )}}{b^{4} c^{2}}\right )} x + \frac{A d^{2}}{b c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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