Optimal. Leaf size=271 \[ -\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac{e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac{3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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Rubi [A] time = 0.349665, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {742, 779, 612, 620, 206} \[ -\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac{e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac{3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 779
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\int (d+e x) \left (\frac{1}{2} d (14 c d-5 b e)+\frac{9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac{e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac{\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{16 c^3}\\ &=\frac{(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}-\frac{\left (3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \sqrt{b x+c x^2} \, dx}{256 c^4}\\ &=-\frac{3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}+\frac{(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac{\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac{3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}+\frac{(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac{\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac{3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}+\frac{(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac{3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.701945, size = 312, normalized size = 1.15 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (28 b^4 c^2 e \left (90 d^2+35 d e x+6 e^2 x^2\right )-16 b^3 c^3 \left (105 d^2 e x+105 d^3+49 d e^2 x^2+9 e^3 x^3\right )+32 b^2 c^4 x \left (42 d^2 e x+35 d^3+21 d e^2 x^2+4 e^3 x^3\right )-210 b^5 c e^2 (7 d+e x)+315 b^6 e^3+128 b c^5 x^2 \left (231 d^2 e x+105 d^3+182 d e^2 x^2+50 e^3 x^3\right )+256 c^6 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )\right )-\frac{105 b^{7/2} \left (-14 b^2 c d e^2+3 b^3 e^3+24 b c^2 d^2 e-16 c^3 d^3\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{35840 c^{11/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 629, normalized size = 2.3 \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 = 2.40864, size = 1586, normalized size = 5.85 \begin{align*} \left [-\frac{105 \,{\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \,{\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \,{\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \,{\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \,{\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \,{\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{71680 \, c^{6}}, -\frac{105 \,{\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \,{\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \,{\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \,{\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \,{\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \,{\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{35840 \, c^{6}}\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 \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34195, size = 493, normalized size = 1.82 \begin{align*} \frac{1}{35840} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c x e^{3} + \frac{14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac{168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac{560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}}{c^{6}}\right )} x + \frac{7 \,{\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )}}{c^{6}}\right )} x - \frac{105 \,{\left (16 \, b^{3} c^{4} d^{3} - 24 \, b^{4} c^{3} d^{2} e + 14 \, b^{5} c^{2} d e^{2} - 3 \, b^{6} c e^{3}\right )}}{c^{6}}\right )} - \frac{3 \,{\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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