Optimal. Leaf size=238 \[ -\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)+B \left (5 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right )}{4 c^{7/2}}+\frac {e \sqrt {b x+c x^2} \left (2 c e x \left (-4 b c (A e+B d)+8 A c^2 d+5 b^2 B e\right )+12 b^2 c e (A e+3 B d)-8 b c^2 d (3 A e+2 B d)+32 A c^3 d^2-15 b^3 B e^2\right )}{4 b^2 c^3} \]
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Rubi [A] time = 0.22, antiderivative size = 238, 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, 779, 620, 206} \begin {gather*} \frac {e \sqrt {b x+c x^2} \left (2 c e x \left (-4 b c (A e+B d)+8 A c^2 d+5 b^2 B e\right )+12 b^2 c e (A e+3 B d)-8 b c^2 d (3 A e+2 B d)+32 A c^3 d^2-15 b^3 B e^2\right )}{4 b^2 c^3}+\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)+B \left (5 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right )}{4 c^{7/2}}-\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 )}{b^2 c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 779
Rule 818
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{3/2}} \, 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 )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \int \frac {(d+e x) \left (\frac {1}{2} b (b B+4 A c) d e+\frac {1}{2} e \left (8 A c^2 d+5 b^2 B e-4 b c (B d+A e)\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{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 )}{b^2 c \sqrt {b x+c x^2}}+\frac {e \left (32 A c^3 d^2-15 b^3 B e^2+12 b^2 c e (3 B d+A e)-8 b c^2 d (2 B d+3 A e)+2 c e \left (8 A c^2 d+5 b^2 B e-4 b c (B d+A e)\right ) x\right ) \sqrt {b x+c x^2}}{4 b^2 c^3}+\frac {\left (3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2-12 b c d e+5 b^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 c^3}\\ &=-\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 )}{b^2 c \sqrt {b x+c x^2}}+\frac {e \left (32 A c^3 d^2-15 b^3 B e^2+12 b^2 c e (3 B d+A e)-8 b c^2 d (2 B d+3 A e)+2 c e \left (8 A c^2 d+5 b^2 B e-4 b c (B d+A e)\right ) x\right ) \sqrt {b x+c x^2}}{4 b^2 c^3}+\frac {\left (3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2-12 b c d e+5 b^2 e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 c^3}\\ &=-\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 )}{b^2 c \sqrt {b x+c x^2}}+\frac {e \left (32 A c^3 d^2-15 b^3 B e^2+12 b^2 c e (3 B d+A e)-8 b c^2 d (2 B d+3 A e)+2 c e \left (8 A c^2 d+5 b^2 B e-4 b c (B d+A e)\right ) x\right ) \sqrt {b x+c x^2}}{4 b^2 c^3}+\frac {3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2-12 b c d e+5 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 229, normalized size = 0.96 \begin {gather*} \frac {3 b^{5/2} e \sqrt {x} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (4 A c e (2 c d-b e)+B \left (5 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right )+\sqrt {c} \left (4 A c \left (3 b^3 e^3 x+b^2 c e^2 x (e x-6 d)-2 b c^2 d^2 (d-3 e x)-4 c^3 d^3 x\right )+b B x \left (-15 b^3 e^3+b^2 c e^2 (36 d-5 e x)+2 b c^2 e \left (-12 d^2+6 d e x+e^2 x^2\right )+8 c^3 d^3\right )\right )}{4 b^2 c^{7/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.79, size = 287, normalized size = 1.21 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (12 A b^3 c e^3 x-24 A b^2 c^2 d e^2 x+4 A b^2 c^2 e^3 x^2-8 A b c^3 d^3+24 A b c^3 d^2 e x-16 A c^4 d^3 x-15 b^4 B e^3 x+36 b^3 B c d e^2 x-5 b^3 B c e^3 x^2-24 b^2 B c^2 d^2 e x+12 b^2 B c^2 d e^2 x^2+2 b^2 B c^2 e^3 x^3+8 b B c^3 d^3 x\right )}{4 b^2 c^3 x (b+c x)}-\frac {3 \log \left (-2 c^{7/2} \sqrt {b x+c x^2}+b c^3+2 c^4 x\right ) \left (-4 A b c e^3+8 A c^2 d e^2+5 b^2 B e^3-12 b B c d e^2+8 B c^2 d^2 e\right )}{8 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 694, normalized size = 2.92 \begin {gather*} \left [\frac {3 \, {\left ({\left (8 \, B b^{2} c^{3} d^{2} e - 4 \, {\left (3 \, B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d e^{2} + {\left (5 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )} e^{3}\right )} x^{2} + {\left (8 \, B b^{3} c^{2} d^{2} e - 4 \, {\left (3 \, B b^{4} c - 2 \, A b^{3} c^{2}\right )} d e^{2} + {\left (5 \, B b^{5} - 4 \, A b^{4} c\right )} e^{3}\right )} x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, B b^{2} c^{3} e^{3} x^{3} - 8 \, A b c^{4} d^{3} + {\left (12 \, B b^{2} c^{3} d e^{2} - {\left (5 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} e^{3}\right )} x^{2} + {\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} - 24 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{2} e + 12 \, {\left (3 \, B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d e^{2} - 3 \, {\left (5 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (b^{2} c^{5} x^{2} + b^{3} c^{4} x\right )}}, -\frac {3 \, {\left ({\left (8 \, B b^{2} c^{3} d^{2} e - 4 \, {\left (3 \, B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d e^{2} + {\left (5 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )} e^{3}\right )} x^{2} + {\left (8 \, B b^{3} c^{2} d^{2} e - 4 \, {\left (3 \, B b^{4} c - 2 \, A b^{3} c^{2}\right )} d e^{2} + {\left (5 \, B b^{5} - 4 \, A b^{4} c\right )} e^{3}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (2 \, B b^{2} c^{3} e^{3} x^{3} - 8 \, A b c^{4} d^{3} + {\left (12 \, B b^{2} c^{3} d e^{2} - {\left (5 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} e^{3}\right )} x^{2} + {\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} - 24 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{2} e + 12 \, {\left (3 \, B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d e^{2} - 3 \, {\left (5 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (b^{2} c^{5} x^{2} + b^{3} c^{4} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 251, normalized size = 1.05 \begin {gather*} -\frac {\frac {8 \, A d^{3}}{b} - {\left ({\left (\frac {2 \, B x e^{3}}{c} + \frac {12 \, B b^{2} c^{2} d e^{2} - 5 \, B b^{3} c e^{3} + 4 \, A b^{2} c^{2} e^{3}}{b^{2} c^{3}}\right )} x + \frac {8 \, B b c^{3} d^{3} - 16 \, A c^{4} d^{3} - 24 \, B b^{2} c^{2} d^{2} e + 24 \, A b c^{3} d^{2} e + 36 \, B b^{3} c d e^{2} - 24 \, A b^{2} c^{2} d e^{2} - 15 \, B b^{4} e^{3} + 12 \, A b^{3} c e^{3}}{b^{2} c^{3}}\right )} x}{4 \, \sqrt {c x^{2} + b x}} - \frac {3 \, {\left (8 \, B c^{2} d^{2} e - 12 \, B b c d e^{2} + 8 \, A c^{2} d e^{2} + 5 \, B b^{2} e^{3} - 4 \, A b c e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 450, normalized size = 1.89 \begin {gather*} \frac {B \,e^{3} x^{3}}{2 \sqrt {c \,x^{2}+b x}\, c}+\frac {A \,e^{3} x^{2}}{\sqrt {c \,x^{2}+b x}\, c}-\frac {5 B b \,e^{3} x^{2}}{4 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {3 B d \,e^{2} x^{2}}{\sqrt {c \,x^{2}+b x}\, c}+\frac {3 A b \,e^{3} x}{\sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {6 A \,d^{2} e x}{\sqrt {c \,x^{2}+b x}\, b}-\frac {6 A d \,e^{2} x}{\sqrt {c \,x^{2}+b x}\, c}-\frac {15 B \,b^{2} e^{3} x}{4 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {9 B b d \,e^{2} x}{\sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {2 B \,d^{3} x}{\sqrt {c \,x^{2}+b x}\, b}-\frac {6 B \,d^{2} e x}{\sqrt {c \,x^{2}+b x}\, c}-\frac {3 A b \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {5}{2}}}+\frac {3 A d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}+\frac {15 B \,b^{2} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {7}{2}}}-\frac {9 B b d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {5}{2}}}+\frac {3 B \,d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}-\frac {2 \left (2 c x +b \right ) A \,d^{3}}{\sqrt {c \,x^{2}+b x}\, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 367, normalized size = 1.54 \begin {gather*} \frac {B e^{3} x^{3}}{2 \, \sqrt {c x^{2} + b x} c} - \frac {5 \, B b e^{3} x^{2}}{4 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, B d^{3} x}{\sqrt {c x^{2} + b x} b} - \frac {4 \, A c d^{3} x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {6 \, A d^{2} e x}{\sqrt {c x^{2} + b x} b} - \frac {15 \, B b^{2} e^{3} x}{4 \, \sqrt {c x^{2} + b x} c^{3}} + \frac {15 \, B b^{2} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {7}{2}}} - \frac {2 \, A d^{3}}{\sqrt {c x^{2} + b x} b} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} x^{2}}{\sqrt {c x^{2} + b x} c} + \frac {3 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b x}{\sqrt {c x^{2} + b x} c^{2}} - \frac {6 \, {\left (B d^{2} e + A d e^{2}\right )} x}{\sqrt {c x^{2} + b x} c} - \frac {3 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {5}{2}}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {3}{2}}} \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 )}^{3/2}} \,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 {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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