Optimal. Leaf size=151 \[ \frac {e \sqrt {b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{b^2 c^2}-\frac {2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{c^{5/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 151, 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, 640, 620, 206} \begin {gather*} \frac {e \sqrt {b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{b^2 c^2}-\frac {2 (d+e x) \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 {e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x) \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 {\frac {1}{2} b (b B+2 A c) d e+\frac {1}{2} e \left (4 A c^2 d+3 b^2 B e-2 b c (B d+A e)\right ) x}{\sqrt {b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac {2 (d+e x) \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 (4 A c^2 d+3 b^2 B e-2 b c (B d+A e)\right ) \sqrt {b x+c x^2}}{b^2 c^2}+\frac {(e (4 B c d-3 b B e+2 A c e)) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c^2}\\ &=-\frac {2 (d+e x) \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 (4 A c^2 d+3 b^2 B e-2 b c (B d+A e)\right ) \sqrt {b x+c x^2}}{b^2 c^2}+\frac {(e (4 B c d-3 b B e+2 A c e)) \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) \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 (4 A c^2 d+3 b^2 B e-2 b c (B d+A e)\right ) \sqrt {b x+c x^2}}{b^2 c^2}+\frac {e (4 B c d-3 b B e+2 A c e) \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 = 0.20, size = 150, normalized size = 0.99 \begin {gather*} \frac {\sqrt {c} \left (b B x \left (3 b^2 e^2+b c e (e x-4 d)+2 c^2 d^2\right )-2 A c \left (b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )\right )-b^{5/2} e \sqrt {x} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) (-2 A c e+3 b B e-4 B c d)}{b^2 c^{5/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 179, normalized size = 1.19 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-2 A b^2 c e^2 x-2 A b c^2 d^2+4 A b c^2 d e x-4 A c^3 d^2 x+3 b^3 B e^2 x-4 b^2 B c d e x+b^2 B c e^2 x^2+2 b B c^2 d^2 x\right )}{b^2 c^2 x (b+c x)}+\frac {\log \left (-2 c^{5/2} \sqrt {b x+c x^2}+b c^2+2 c^3 x\right ) \left (-2 A c e^2+3 b B e^2-4 B c d e\right )}{2 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 446, normalized size = 2.95 \begin {gather*} \left [\frac {{\left ({\left (4 \, B b^{2} c^{2} d e - {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, B b^{3} c d e - {\left (3 \, B b^{4} - 2 \, A b^{3} c\right )} e^{2}\right )} x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (B b^{2} c^{2} e^{2} x^{2} - 2 \, A b c^{3} d^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d e + {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (b^{2} c^{4} x^{2} + b^{3} c^{3} x\right )}}, -\frac {{\left ({\left (4 \, B b^{2} c^{2} d e - {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, B b^{3} c d e - {\left (3 \, B b^{4} - 2 \, A b^{3} c\right )} e^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (B b^{2} c^{2} e^{2} x^{2} - 2 \, A b c^{3} d^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d e + {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{b^{2} c^{4} x^{2} + b^{3} c^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 155, normalized size = 1.03 \begin {gather*} -\frac {\frac {2 \, A d^{2}}{b} - {\left (\frac {B x e^{2}}{c} + \frac {2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 4 \, A b c^{2} d e + 3 \, B b^{3} e^{2} - 2 \, A b^{2} c e^{2}}{b^{2} c^{2}}\right )} x}{\sqrt {c x^{2} + b x}} - \frac {{\left (4 \, B c d e - 3 \, B b e^{2} + 2 \, A c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 252, normalized size = 1.67 \begin {gather*} \frac {B \,e^{2} x^{2}}{\sqrt {c \,x^{2}+b x}\, c}+\frac {4 A d e x}{\sqrt {c \,x^{2}+b x}\, b}-\frac {2 A \,e^{2} x}{\sqrt {c \,x^{2}+b x}\, c}+\frac {3 B b \,e^{2} x}{\sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {2 B \,d^{2} x}{\sqrt {c \,x^{2}+b x}\, b}-\frac {4 B d e x}{\sqrt {c \,x^{2}+b x}\, c}+\frac {A \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}-\frac {3 B b \,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 {2 B d 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^{2}}{\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 = 0.59, size = 227, normalized size = 1.50 \begin {gather*} \frac {B e^{2} x^{2}}{\sqrt {c x^{2} + b x} c} + \frac {2 \, B d^{2} x}{\sqrt {c x^{2} + b x} b} - \frac {4 \, A c d^{2} x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {4 \, A d e x}{\sqrt {c x^{2} + b x} b} + \frac {3 \, B b e^{2} x}{\sqrt {c x^{2} + b x} c^{2}} - \frac {3 \, B b e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {5}{2}}} - \frac {2 \, A d^{2}}{\sqrt {c x^{2} + b x} b} - \frac {2 \, {\left (2 \, B d e + A e^{2}\right )} x}{\sqrt {c x^{2} + b x} c} + \frac {{\left (2 \, B d e + A 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 = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2}{{\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 )^{2}}{\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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