Optimal. Leaf size=423 \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac{\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac{B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]
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Rubi [A] time = 0.433299, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {832, 779, 612, 620, 206} \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac{\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac{B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx &=\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\int (d+e x) \left (-\frac{1}{2} (7 b B-18 A c) d+\frac{1}{2} (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=\frac{\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=-\frac{5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt{b x+c x^2} \, dx}{8192 c^5}\\ &=\frac{5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{32768 c^6}-\frac{5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{65536 c^6}\\ &=\frac{5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{32768 c^6}-\frac{5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{32768 c^6}\\ &=\frac{5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{32768 c^6}-\frac{5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end{align*}
Mathematica [A] time = 1.50674, size = 417, normalized size = 0.99 \[ \frac{(x (b+c x))^{7/2} \left (\frac{1323 A \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \left (\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (8 b^3 c^2 x^2+432 b^2 c^3 x^3-10 b^4 c x+15 b^5+640 b c^4 x^4+256 c^5 x^5\right )-15 b^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{1024 c^{9/2} x^{7/2} \sqrt{\frac{c x}{b}+1}}+\frac{5103 A e (b+c x)^3 (2 c d-b e)}{c}+7938 A e (b+c x)^3 (d+e x)+\frac{189 B \left (11 b^2 e^2-36 b c d e+32 c^2 d^2\right ) \left (\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (-56 b^4 c^2 x^2+48 b^3 c^3 x^3+4736 b^2 c^4 x^4+70 b^5 c x-105 b^6+7424 b c^5 x^5+3072 c^6 x^6\right )+105 b^{13/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{2048 c^{11/2} x^{7/2} \sqrt{\frac{c x}{b}+1}}+\frac{441 B e x (b+c x)^3 (20 c d-11 b e)}{c}+7056 B e x (b+c x)^3 (d+e x)\right )}{63504 c (b+c x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 1227, normalized size = 2.9 \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 = 1.75418, size = 2921, normalized size = 6.91 \begin{align*} \text{result too large to display} \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{5}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46457, size = 919, normalized size = 2.17 \begin{align*} \frac{1}{2064384} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, B c^{2} x e^{2} + \frac{36 \, B c^{10} d e + 37 \, B b c^{9} e^{2} + 18 \, A c^{10} e^{2}}{c^{8}}\right )} x + \frac{3 \,{\left (96 \, B c^{10} d^{2} + 396 \, B b c^{9} d e + 192 \, A c^{10} d e + 103 \, B b^{2} c^{8} e^{2} + 198 \, A b c^{9} e^{2}\right )}}{c^{8}}\right )} x + \frac{2784 \, B b c^{9} d^{2} + 1344 \, A c^{10} d^{2} + 2916 \, B b^{2} c^{8} d e + 5568 \, A b c^{9} d e + 5 \, B b^{3} c^{7} e^{2} + 1458 \, A b^{2} c^{8} e^{2}}{c^{8}}\right )} x + \frac{3552 \, B b^{2} c^{8} d^{2} + 6720 \, A b c^{9} d^{2} + 36 \, B b^{3} c^{7} d e + 7104 \, A b^{2} c^{8} d e - 11 \, B b^{4} c^{6} e^{2} + 18 \, A b^{3} c^{7} e^{2}}{c^{8}}\right )} x + \frac{9 \,{\left (32 \, B b^{3} c^{7} d^{2} + 4032 \, A b^{2} c^{8} d^{2} - 36 \, B b^{4} c^{6} d e + 64 \, A b^{3} c^{7} d e + 11 \, B b^{5} c^{5} e^{2} - 18 \, A b^{4} c^{6} e^{2}\right )}}{c^{8}}\right )} x - \frac{21 \,{\left (32 \, B b^{4} c^{6} d^{2} - 64 \, A b^{3} c^{7} d^{2} - 36 \, B b^{5} c^{5} d e + 64 \, A b^{4} c^{6} d e + 11 \, B b^{6} c^{4} e^{2} - 18 \, A b^{5} c^{5} e^{2}\right )}}{c^{8}}\right )} x + \frac{105 \,{\left (32 \, B b^{5} c^{5} d^{2} - 64 \, A b^{4} c^{6} d^{2} - 36 \, B b^{6} c^{4} d e + 64 \, A b^{5} c^{5} d e + 11 \, B b^{7} c^{3} e^{2} - 18 \, A b^{6} c^{4} e^{2}\right )}}{c^{8}}\right )} x - \frac{315 \,{\left (32 \, B b^{6} c^{4} d^{2} - 64 \, A b^{5} c^{5} d^{2} - 36 \, B b^{7} c^{3} d e + 64 \, A b^{6} c^{4} d e + 11 \, B b^{8} c^{2} e^{2} - 18 \, A b^{7} c^{3} e^{2}\right )}}{c^{8}}\right )} - \frac{5 \,{\left (32 \, B b^{7} c^{2} d^{2} - 64 \, A b^{6} c^{3} d^{2} - 36 \, B b^{8} c d e + 64 \, A b^{7} c^{2} d e + 11 \, B b^{9} e^{2} - 18 \, A b^{8} c e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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