Optimal. Leaf size=305 \[ \frac{\sqrt{b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (360 b^2 c d e^2-105 b^3 e^3-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (144 b^2 c^2 d e (A e+B d)-40 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac{\sqrt{b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac{B \sqrt{b x+c x^2} (d+e x)^3}{4 c} \]
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Rubi [A] time = 0.442533, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {832, 779, 620, 206} \[ \frac{\sqrt{b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (360 b^2 c d e^2-105 b^3 e^3-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (144 b^2 c^2 d e (A e+B d)-40 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac{\sqrt{b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac{B \sqrt{b x+c x^2} (d+e x)^3}{4 c} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^3}{\sqrt{b x+c x^2}} \, dx &=\frac{B (d+e x)^3 \sqrt{b x+c x^2}}{4 c}+\frac{\int \frac{(d+e x)^2 \left (-\frac{1}{2} (b B-8 A c) d+\frac{1}{2} (6 B c d-7 b B e+8 A c e) x\right )}{\sqrt{b x+c x^2}} \, dx}{4 c}\\ &=\frac{(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{B (d+e x)^3 \sqrt{b x+c x^2}}{4 c}+\frac{\int \frac{(d+e x) \left (-\frac{1}{4} d \left (12 b B c d-48 A c^2 d-7 b^2 B e+8 A b c e\right )+\frac{1}{4} \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right )}{\sqrt{b x+c x^2}} \, dx}{12 c^2}\\ &=\frac{(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{B (d+e x)^3 \sqrt{b x+c x^2}}{4 c}+\frac{\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{192 c^4}+\frac{\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^4}\\ &=\frac{(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{B (d+e x)^3 \sqrt{b x+c x^2}}{4 c}+\frac{\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{192 c^4}+\frac{\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c^4}\\ &=\frac{(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{B (d+e x)^3 \sqrt{b x+c x^2}}{4 c}+\frac{\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{192 c^4}+\frac{\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.824778, size = 278, normalized size = 0.91 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (8 A c e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (10 b^2 c e^2 (36 d+7 e x)-105 b^3 e^3-8 b c^2 e \left (54 d^2+30 d e x+7 e^2 x^2\right )+48 c^3 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right )\right )+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) \left (144 b^2 c^2 d e (A e+B d)-40 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 646, normalized size = 2.1 \begin{align*} \text{result too large to display} \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 = 1.94792, size = 1370, normalized size = 4.49 \begin{align*} \left [\frac{3 \,{\left (64 \,{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \,{\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \,{\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \,{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \,{\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 72 \,{\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \,{\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} e^{3} + 8 \,{\left (24 \, B c^{4} d e^{2} -{\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \,{\left (144 \, B c^{4} d^{2} e - 24 \,{\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + 5 \,{\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{384 \, c^{5}}, \frac{3 \,{\left (64 \,{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \,{\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \,{\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \,{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \,{\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 72 \,{\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \,{\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} e^{3} + 8 \,{\left (24 \, B c^{4} d e^{2} -{\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \,{\left (144 \, B c^{4} d^{2} e - 24 \,{\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + 5 \,{\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{192 \, c^{5}}\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 )^{3}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46025, size = 420, normalized size = 1.38 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (\frac{6 \, B x e^{3}}{c} + \frac{24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac{144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac{3 \,{\left (64 \, B c^{3} d^{3} - 144 \, B b c^{2} d^{2} e + 192 \, A c^{3} d^{2} e + 120 \, B b^{2} c d e^{2} - 144 \, A b c^{2} d e^{2} - 35 \, B b^{3} e^{3} + 40 \, A b^{2} c e^{3}\right )}}{c^{4}}\right )} + \frac{{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 144 \, A b^{2} c^{2} d e^{2} - 35 \, B b^{4} e^{3} + 40 \, A b^{3} c e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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