Optimal. Leaf size=209 \[ -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \]
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Rubi [A] time = 0.19, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {779, 612, 620, 206} \begin {gather*} -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 612
Rule 620
Rule 779
Rubi steps
\begin {align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx &=-\frac {(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (\frac {7}{2} b^2 B e+6 c (2 A c d-b (B d+A e))\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac {\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}-\frac {\left (b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+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 )}{512 c^4}\\ &=-\frac {b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 245, normalized size = 1.17 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {15 b^{7/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (10 b^4 c (18 A e+18 B d+7 B e x)-8 b^3 c^2 (15 A (3 d+e x)+B x (15 d+7 e x))+48 b^2 c^3 x (A (5 d+2 e x)+B x (2 d+e x))+64 b c^4 x^2 (A (45 d+33 e x)+B x (33 d+26 e x))+128 c^5 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))-105 b^5 B e\right )\right )}{7680 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.27, size = 303, normalized size = 1.45 \begin {gather*} \frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (12 A b^5 c e-24 A b^4 c^2 d-7 b^6 B e+12 b^5 B c d\right )}{1024 c^{9/2}}+\frac {\sqrt {b x+c x^2} \left (180 A b^4 c e-360 A b^3 c^2 d-120 A b^3 c^2 e x+240 A b^2 c^3 d x+96 A b^2 c^3 e x^2+2880 A b c^4 d x^2+2112 A b c^4 e x^3+1920 A c^5 d x^3+1536 A c^5 e x^4-105 b^5 B e+180 b^4 B c d+70 b^4 B c e x-120 b^3 B c^2 d x-56 b^3 B c^2 e x^2+96 b^2 B c^3 d x^2+48 b^2 B c^3 e x^3+2112 b B c^4 d x^3+1664 b B c^4 e x^4+1536 B c^5 d x^4+1280 B c^5 e x^5\right )}{7680 c^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 603, normalized size = 2.89 \begin {gather*} \left [\frac {15 \, {\left (12 \, {\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d - {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (1280 \, B c^{6} e x^{5} + 128 \, {\left (12 \, B c^{6} d + {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \, {\left (4 \, {\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d + {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (12 \, {\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d - {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \, {\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \, {\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \, {\left (12 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d - {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, \frac {15 \, {\left (12 \, {\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d - {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (1280 \, B c^{6} e x^{5} + 128 \, {\left (12 \, B c^{6} d + {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \, {\left (4 \, {\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d + {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (12 \, {\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d - {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \, {\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \, {\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \, {\left (12 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d - {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 317, normalized size = 1.52 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B c x e + \frac {12 \, B c^{6} d + 13 \, B b c^{5} e + 12 \, A c^{6} e}{c^{5}}\right )} x + \frac {3 \, {\left (44 \, B b c^{5} d + 40 \, A c^{6} d + B b^{2} c^{4} e + 44 \, A b c^{5} e\right )}}{c^{5}}\right )} x + \frac {12 \, B b^{2} c^{4} d + 360 \, A b c^{5} d - 7 \, B b^{3} c^{3} e + 12 \, A b^{2} c^{4} e}{c^{5}}\right )} x - \frac {5 \, {\left (12 \, B b^{3} c^{3} d - 24 \, A b^{2} c^{4} d - 7 \, B b^{4} c^{2} e + 12 \, A b^{3} c^{3} e\right )}}{c^{5}}\right )} x + \frac {15 \, {\left (12 \, B b^{4} c^{2} d - 24 \, A b^{3} c^{3} d - 7 \, B b^{5} c e + 12 \, A b^{4} c^{2} e\right )}}{c^{5}}\right )} + \frac {{\left (12 \, B b^{5} c d - 24 \, A b^{4} c^{2} d - 7 \, B b^{6} e + 12 \, A b^{5} c e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 544, normalized size = 2.60 \begin {gather*} -\frac {3 A \,b^{5} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}+\frac {3 A \,b^{4} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {5}{2}}}+\frac {7 B \,b^{6} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {3 B \,b^{5} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} e x}{64 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{2} d x}{32 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{4} e x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,b^{3} d x}{64 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{4} e}{128 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} d}{64 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b e x}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A d x}{4}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{5} e}{512 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,b^{4} d}{128 c^{3}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} e x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b d x}{8 c}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} e}{16 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b d}{8 c}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} e}{192 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} d}{16 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B e x}{6 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A e}{5 c}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b e}{60 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B d}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 434, normalized size = 2.08 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A d x - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2} d x}{32 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{4} e x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} e x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B e x}{6 \, c} + \frac {3 \, A b^{4} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} + \frac {7 \, B b^{6} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} d}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b d}{8 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{5} e}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b e}{60 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{3} x}{64 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b x}{8 \, c} - \frac {3 \, {\left (B d + A e\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} + \frac {3 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{4}}{128 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b^{2}}{16 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B d + A e\right )}}{5 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right ) \left (d + e x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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