Optimal. Leaf size=124 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.0641323, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx\\ &=-\frac{1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac{(5 e) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)}\\ &=-\frac{5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac{1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac{(5 b e) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^2}\\ &=-\frac{5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac{1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac{5 b e}{(b d-a e)^3 \sqrt{d+e x}}-\frac{\left (5 b^2 e\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 (b d-a e)^3}\\ &=-\frac{5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac{1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac{5 b e}{(b d-a e)^3 \sqrt{d+e x}}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{(b d-a e)^3}\\ &=-\frac{5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac{1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac{5 b e}{(b d-a e)^3 \sqrt{d+e x}}+\frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0149508, size = 50, normalized size = 0.4 \[ -\frac{2 e \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.204, size = 125, normalized size = 1. \begin{align*} -{\frac{2\,e}{3\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{be}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) }\sqrt{ex+d}}+5\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\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 = 2.02, size = 1592, normalized size = 12.84 \begin{align*} \left [-\frac{15 \,{\left (b^{2} e^{3} x^{3} + a b d^{2} e +{\left (2 \, b^{2} d e^{2} + a b e^{3}\right )} x^{2} +{\left (b^{2} d^{2} e + 2 \, a b d e^{2}\right )} x\right )} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 2 \,{\left (15 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 14 \, a b d e - 2 \, a^{2} e^{2} + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}}, \frac{15 \,{\left (b^{2} e^{3} x^{3} + a b d^{2} e +{\left (2 \, b^{2} d e^{2} + a b e^{3}\right )} x^{2} +{\left (b^{2} d^{2} e + 2 \, a b d e^{2}\right )} x\right )} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}}}{b e x + b d}\right ) -{\left (15 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 14 \, a b d e - 2 \, a^{2} e^{2} + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}}\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{1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24007, size = 302, normalized size = 2.44 \begin{align*} -\frac{5 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{\sqrt{x e + d} b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} b e + b d e - a e^{2}\right )}}{3 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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