Optimal. Leaf size=85 \[ \frac {2 B e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {777, 620, 206} \begin {gather*} \frac {2 B e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}-\frac {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 777
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {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 {(B e) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c}\\ &=-\frac {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 B e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c}\\ &=-\frac {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 B e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 101, normalized size = 1.19 \begin {gather*} \frac {2 b^{5/2} B e \sqrt {x} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )-2 \sqrt {c} (A c (b d-b e x+2 c d x)+b B x (b e-c d))}{b^2 c^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 107, normalized size = 1.26 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (A b c d-A b c e x+2 A c^2 d x+b^2 B e x-b B c d x\right )}{b^2 c x (b+c x)}-\frac {B e \log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 254, normalized size = 2.99 \begin {gather*} \left [\frac {{\left (B b^{2} c e x^{2} + B b^{3} e x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (A b c^{2} d - {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - A b c^{2}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{b^{2} c^{3} x^{2} + b^{3} c^{2} x}, -\frac {2 \, {\left ({\left (B b^{2} c e x^{2} + B b^{3} e x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (A b c^{2} d - {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - A b c^{2}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{b^{2} c^{3} x^{2} + b^{3} c^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 95, normalized size = 1.12 \begin {gather*} -\frac {B e \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left (\frac {A d}{b} - \frac {{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + A b c e\right )} x}{b^{2} c}\right )}}{\sqrt {c x^{2} + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 113, normalized size = 1.33 \begin {gather*} \frac {2 A e x}{\sqrt {c \,x^{2}+b x}\, b}+\frac {2 B d x}{\sqrt {c \,x^{2}+b x}\, b}-\frac {2 B e x}{\sqrt {c \,x^{2}+b x}\, c}+\frac {B 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}{\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 = 0.57, size = 125, normalized size = 1.47 \begin {gather*} \frac {2 \, B d x}{\sqrt {c x^{2} + b x} b} - \frac {4 \, A c d x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, A e x}{\sqrt {c x^{2} + b x} b} - \frac {2 \, B e x}{\sqrt {c x^{2} + b x} c} + \frac {B e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {3}{2}}} - \frac {2 \, A d}{\sqrt {c x^{2} + b x} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.27, size = 101, normalized size = 1.19 \begin {gather*} \frac {B\,e\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{c^{3/2}}-\frac {2\,A\,b\,d-2\,A\,b\,e\,x+4\,A\,c\,d\,x}{b^2\,\sqrt {c\,x^2+b\,x}}+\frac {2\,B\,d\,x}{b\,\sqrt {x\,\left (b+c\,x\right )}}-\frac {2\,B\,e\,x}{c\,\sqrt {c\,x^2+b\,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 )}{\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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