Optimal. Leaf size=345 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac{\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac{B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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Rubi [A] time = 0.336663, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac{\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac{B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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 )^{3/2} \, dx &=\frac{B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\int (d+e x) \left (-\frac{1}{2} (5 b B-14 A c) d+\frac{1}{2} (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac{B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac{\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=\frac{\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}-\frac{\left (b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt{b x+c x^2} \, dx}{256 c^4}\\ &=-\frac{b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}+\frac{\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac{\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac{b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}+\frac{\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac{\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 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 )}{1024 c^5}\\ &=-\frac{b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}+\frac{\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac{b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 1.66677, size = 415, normalized size = 1.2 \[ \frac{\sqrt{x (b+c x)} \left (14 A c \left (\frac{5}{4} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (b c x \sqrt{\frac{c x}{b}+1} \left (2 b^2 c x-3 b^3+24 b c^2 x^2+16 c^3 x^3\right )+3 b^{9/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )+320 b c^4 e x^3 (b+c x)^2 \sqrt{\frac{c x}{b}+1} (d+e x)-224 b c^3 e x^3 (b+c x)^2 \sqrt{\frac{c x}{b}+1} (b e-2 c d)\right )+B \left (\frac{7}{4} \left (9 b^2 e^2-28 b c d e+24 c^2 d^2\right ) \left (b c x \sqrt{\frac{c x}{b}+1} \left (8 b^2 c^2 x^2-10 b^3 c x+15 b^4+176 b c^3 x^3+128 c^4 x^4\right )-15 b^{11/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )+3840 b c^5 e x^4 (b+c x)^2 \sqrt{\frac{c x}{b}+1} (d+e x)+320 b c^4 e x^4 (b+c x)^2 \sqrt{\frac{c x}{b}+1} (16 c d-9 b e)\right )\right )}{26880 b c^6 x \sqrt{\frac{c x}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 949, normalized size = 2.8 \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 = 1.13538, size = 2209, normalized size = 6.4 \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{3}{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 = 1.21168, size = 699, normalized size = 2.03 \begin{align*} \frac{1}{107520} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B c x e^{2} + \frac{28 \, B c^{7} d e + 15 \, B b c^{6} e^{2} + 14 \, A c^{7} e^{2}}{c^{6}}\right )} x + \frac{168 \, B c^{7} d^{2} + 364 \, B b c^{6} d e + 336 \, A c^{7} d e + 3 \, B b^{2} c^{5} e^{2} + 182 \, A b c^{6} e^{2}}{c^{6}}\right )} x + \frac{3 \,{\left (616 \, B b c^{6} d^{2} + 560 \, A c^{7} d^{2} + 28 \, B b^{2} c^{5} d e + 1232 \, A b c^{6} d e - 9 \, B b^{3} c^{4} e^{2} + 14 \, A b^{2} c^{5} e^{2}\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (24 \, B b^{2} c^{5} d^{2} + 720 \, A b c^{6} d^{2} - 28 \, B b^{3} c^{4} d e + 48 \, A b^{2} c^{5} d e + 9 \, B b^{4} c^{3} e^{2} - 14 \, A b^{3} c^{4} e^{2}\right )}}{c^{6}}\right )} x - \frac{35 \,{\left (24 \, B b^{3} c^{4} d^{2} - 48 \, A b^{2} c^{5} d^{2} - 28 \, B b^{4} c^{3} d e + 48 \, A b^{3} c^{4} d e + 9 \, B b^{5} c^{2} e^{2} - 14 \, A b^{4} c^{3} e^{2}\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (24 \, B b^{4} c^{3} d^{2} - 48 \, A b^{3} c^{4} d^{2} - 28 \, B b^{5} c^{2} d e + 48 \, A b^{4} c^{3} d e + 9 \, B b^{6} c e^{2} - 14 \, A b^{5} c^{2} e^{2}\right )}}{c^{6}}\right )} + \frac{{\left (24 \, B b^{5} c^{2} d^{2} - 48 \, A b^{4} c^{3} d^{2} - 28 \, B b^{6} c d e + 48 \, A b^{5} c^{2} d e + 9 \, B b^{7} e^{2} - 14 \, A b^{6} c e^{2}\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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