Optimal. Leaf size=220 \[ -\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {2 B e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {818, 777, 620, 206} \begin {gather*} \frac {2 \left (x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3+2 b^3 B c d e^2-3 b^4 B e^3\right )+b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 B e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 777
Rule 818
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x) \left (-\frac {1}{2} d \left (8 A c^2 d+b^2 B e-4 b c (B d+2 A e)\right )+\frac {3}{2} b^2 B e^2 x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c}\\ &=-\frac {2 (d+e x)^2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b c d^2 \left (8 A c^2 d+b^2 B e-4 b c (B d+2 A e)\right )+\left (16 A c^4 d^3+2 b^3 B c d e^2-3 b^4 B e^3-8 b c^3 d^2 (B d+3 A e)+2 b^2 c^2 d e (3 B d+4 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (B e^3\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c^2}\\ &=-\frac {2 (d+e x)^2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b c d^2 \left (8 A c^2 d+b^2 B e-4 b c (B d+2 A e)\right )+\left (16 A c^4 d^3+2 b^3 B c d e^2-3 b^4 B e^3-8 b c^3 d^2 (B d+3 A e)+2 b^2 c^2 d e (3 B d+4 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (2 B e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^2}\\ &=-\frac {2 (d+e x)^2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b c d^2 \left (8 A c^2 d+b^2 B e-4 b c (B d+2 A e)\right )+\left (16 A c^4 d^3+2 b^3 B c d e^2-3 b^4 B e^3-8 b c^3 d^2 (B d+3 A e)+2 b^2 c^2 d e (3 B d+4 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {2 B e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 5.48, size = 199, normalized size = 0.90 \begin {gather*} \frac {x^{5/2} \left (\frac {2 B e^3 (b+c x)^{5/2} \log \left (\sqrt {c} \sqrt {b+c x}+c \sqrt {x}\right )}{c^{5/2}}-\frac {2 (b+c x) \left (x^2 (b+c x) (c d-b e)^2 \left (b c (5 B d-A e)-8 A c^2 d+4 b^2 B e\right )+c^2 d^2 x (b+c x)^2 (9 A b e-8 A c d+3 b B d)+b x^2 (b B-A c) (c d-b e)^3+A b c^2 d^3 (b+c x)^2\right )}{3 b^4 c^2 x^{3/2}}\right )}{(x (b+c x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.96, size = 333, normalized size = 1.51 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (A b^3 c^2 d^3+9 A b^3 c^2 d^2 e x-9 A b^3 c^2 d e^2 x^2-A b^3 c^2 e^3 x^3-6 A b^2 c^3 d^3 x+36 A b^2 c^3 d^2 e x^2-6 A b^2 c^3 d e^2 x^3-24 A b c^4 d^3 x^2+24 A b c^4 d^2 e x^3-16 A c^5 d^3 x^3+3 b^5 B e^3 x^2+4 b^4 B c e^3 x^3+3 b^3 B c^2 d^3 x-9 b^3 B c^2 d^2 e x^2-3 b^3 B c^2 d e^2 x^3+12 b^2 B c^3 d^3 x^2-6 b^2 B c^3 d^2 e x^3+8 b B c^4 d^3 x^3\right )}{3 b^4 c^2 x^2 (b+c x)^2}-\frac {B e^3 \log \left (-2 c^{5/2} \sqrt {b x+c x^2}+b c^2+2 c^3 x\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 671, normalized size = 3.05 \begin {gather*} \left [\frac {3 \, {\left (B b^{4} c^{2} e^{3} x^{4} + 2 \, B b^{5} c e^{3} x^{3} + B b^{6} e^{3} x^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (A b^{3} c^{3} d^{3} + {\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} - 6 \, {\left (B b^{2} c^{4} - 4 \, A b c^{5}\right )} d^{2} e - 3 \, {\left (B b^{3} c^{3} + 2 \, A b^{2} c^{4}\right )} d e^{2} + {\left (4 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 4 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{2} + 3 \, {\left (3 \, A b^{3} c^{3} d^{2} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (B b^{4} c^{2} e^{3} x^{4} + 2 \, B b^{5} c e^{3} x^{3} + B b^{6} e^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (A b^{3} c^{3} d^{3} + {\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} - 6 \, {\left (B b^{2} c^{4} - 4 \, A b c^{5}\right )} d^{2} e - 3 \, {\left (B b^{3} c^{3} + 2 \, A b^{2} c^{4}\right )} d e^{2} + {\left (4 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 4 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{2} + 3 \, {\left (3 \, A b^{3} c^{3} d^{2} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3}\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 289, normalized size = 1.31 \begin {gather*} -\frac {B e^{3} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {5}{2}}} - \frac {2 \, {\left (\frac {A d^{3}}{b} + {\left (x {\left (\frac {{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} + 4 \, B b^{4} c e^{3} - A b^{3} c^{2} e^{3}\right )} x}{b^{4} c^{2}} + \frac {3 \, {\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e - 3 \, A b^{3} c^{2} d e^{2} + B b^{5} e^{3}\right )}}{b^{4} c^{2}}\right )} + \frac {3 \, {\left (B b^{3} c^{2} d^{3} - 2 \, A b^{2} c^{3} d^{3} + 3 \, A b^{3} c^{2} d^{2} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 680, normalized size = 3.09 \begin {gather*} -\frac {B \,e^{3} x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {A \,e^{3} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {B b \,e^{3} x^{2}}{2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {3 B d \,e^{2} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {A b \,e^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {2 A \,d^{2} e x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}-\frac {4 A c \,d^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2}}-\frac {2 A d \,e^{2} x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {B \,b^{2} e^{3} x}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}-\frac {B b d \,e^{2} x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {2 B \,d^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}-\frac {2 B \,d^{2} e x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {2 A \,e^{3} x}{3 \sqrt {c \,x^{2}+b x}\, b c}-\frac {2 A \,d^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}+\frac {4 A d \,e^{2} x}{\sqrt {c \,x^{2}+b x}\, b^{2}}-\frac {16 A c \,d^{2} e x}{\sqrt {c \,x^{2}+b x}\, b^{3}}+\frac {32 A \,c^{2} d^{3} x}{3 \sqrt {c \,x^{2}+b x}\, b^{4}}+\frac {2 B d \,e^{2} x}{\sqrt {c \,x^{2}+b x}\, b c}+\frac {4 B \,d^{2} e x}{\sqrt {c \,x^{2}+b x}\, b^{2}}-\frac {16 B c \,d^{3} x}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}-\frac {7 B \,e^{3} x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {B \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}+\frac {2 A d \,e^{2}}{\sqrt {c \,x^{2}+b x}\, b c}-\frac {8 A \,d^{2} e}{\sqrt {c \,x^{2}+b x}\, b^{2}}+\frac {16 A c \,d^{3}}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}+\frac {A \,e^{3}}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}-\frac {B b \,e^{3}}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {2 B \,d^{2} e}{\sqrt {c \,x^{2}+b x}\, b c}-\frac {8 B \,d^{3}}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}+\frac {B d \,e^{2}}{\sqrt {c \,x^{2}+b x}\, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 550, normalized size = 2.50 \begin {gather*} -\frac {1}{3} \, B e^{3} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {4 \, A c d^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{3} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {4 \, B e^{3} x}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {B e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {2 \, A d^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{3}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B e^{3}}{3 \, b c^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, {\left (B d^{2} e + A d e^{2}\right )} x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{3} + 3 \, A d^{2} e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, {\left (B d^{2} e + A d e^{2}\right )} x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {2 \, {\left (3 \, B d e^{2} + A e^{3}\right )} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {16 \, {\left (B d^{3} + 3 \, A d^{2} e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d^{3} + 3 \, A d^{2} e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {3 \, B d e^{2} + A e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, {\left (B d^{2} e + A d e^{2}\right )}}{\sqrt {c x^{2} + b x} b 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 \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,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 \frac {\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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