Optimal. Leaf size=140 \[ -\frac{A b-a B}{b (a+b x) \sqrt{d+e x} (b d-a e)}+\frac{a B e-3 A b e+2 b B d}{b \sqrt{d+e x} (b d-a e)^2}-\frac{(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{5/2}} \]
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Rubi [A] time = 0.145376, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \[ -\frac{A b-a B}{b (a+b x) \sqrt{d+e x} (b d-a e)}+\frac{a B e-3 A b e+2 b B d}{b \sqrt{d+e x} (b d-a e)^2}-\frac{(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{(a+b x)^2 (d+e x)^{3/2}} \, dx\\ &=-\frac{A b-a B}{b (b d-a e) (a+b x) \sqrt{d+e x}}+\frac{(2 b B d-3 A b e+a B e) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 b (b d-a e)}\\ &=\frac{2 b B d-3 A b e+a B e}{b (b d-a e)^2 \sqrt{d+e x}}-\frac{A b-a B}{b (b d-a e) (a+b x) \sqrt{d+e x}}+\frac{(2 b B d-3 A b e+a B e) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 (b d-a e)^2}\\ &=\frac{2 b B d-3 A b e+a B e}{b (b d-a e)^2 \sqrt{d+e x}}-\frac{A b-a B}{b (b d-a e) (a+b x) \sqrt{d+e x}}+\frac{(2 b B d-3 A b e+a B e) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e (b d-a e)^2}\\ &=\frac{2 b B d-3 A b e+a B e}{b (b d-a e)^2 \sqrt{d+e x}}-\frac{A b-a B}{b (b d-a e) (a+b x) \sqrt{d+e x}}-\frac{(2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0427089, size = 95, normalized size = 0.68 \[ \frac{(a+b x) (a B e-3 A b e+2 b B d) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )-(A b-a B) (b d-a e)}{b (a+b x) \sqrt{d+e x} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 253, normalized size = 1.8 \begin{align*} -2\,{\frac{Ae}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{Bd}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}-{\frac{Abe}{ \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{aBe}{ \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}-3\,{\frac{Abe}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+{\frac{aBe}{ \left ( ae-bd \right ) ^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+2\,{\frac{Bbd}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \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 = 1.3093, size = 1601, normalized size = 11.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14404, size = 275, normalized size = 1.96 \begin{align*} \frac{{\left (2 \, B b d + B a e - 3 \, A b e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (x e + d\right )} B b d - 2 \, B b d^{2} +{\left (x e + d\right )} B a e - 3 \,{\left (x e + d\right )} A b e + 2 \, B a d e + 2 \, A b d e - 2 \, A a e^{2}}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{\frac{3}{2}} b - \sqrt{x e + d} b d + \sqrt{x e + d} a e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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