Optimal. Leaf size=126 \[ \frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{B e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]
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Rubi [A] time = 0.0642382, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {777, 621, 206} \[ \frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{B e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 777
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{(B e) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c}\\ &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{(2 B e) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c}\\ &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{B e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.409841, size = 127, normalized size = 1.01 \[ \frac{\frac{2 \sqrt{c} (A c (-2 a e+b (d-e x)+2 c d x)+B (a b e-2 a c (d+e x)+b x (b e-c d)))}{\sqrt{a+x (b+c x)}}-B e \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 341, normalized size = 2.7 \begin{align*} -{\frac{Bex}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{bBe}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{2}Bex}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{3}Be}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{Be\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{Ae}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{Bd}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{Abex}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{Bbdx}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{A{b}^{2}e}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{B{b}^{2}d}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+2\,{\frac{Ad \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 11.0751, size = 1060, normalized size = 8.41 \begin{align*} \left [\frac{{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} e x^{2} +{\left (B b^{3} - 4 \, B a b c\right )} e x +{\left (B a b^{2} - 4 \, B a^{2} c\right )} e\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left ({\left (2 \, B a - A b\right )} c^{2} d -{\left (B a b c - 2 \, A a c^{2}\right )} e +{\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d -{\left (B b^{2} c -{\left (2 \, B a + A b\right )} c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac{{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} e x^{2} +{\left (B b^{3} - 4 \, B a b c\right )} e x +{\left (B a b^{2} - 4 \, B a^{2} c\right )} e\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left ({\left (2 \, B a - A b\right )} c^{2} d -{\left (B a b c - 2 \, A a c^{2}\right )} e +{\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d -{\left (B b^{2} c -{\left (2 \, B a + A b\right )} c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (d + e x\right )}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29317, size = 198, normalized size = 1.57 \begin{align*} -\frac{B e \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} + \frac{2 \,{\left (\frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{2 \, B a c d - A b c d - B a b e + 2 \, A a c e}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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