Optimal. Leaf size=167 \[ -\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac{35 b e^2}{4 \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.0827507, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \[ -\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac{35 b e^2}{4 \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^3 (d+e x)^{5/2}} \, dx\\ &=-\frac{1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac{(7 e) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{4 (b d-a e)}\\ &=-\frac{1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac{\left (35 e^2\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^2}\\ &=\frac{35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac{1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac{\left (35 b e^2\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3}\\ &=\frac{35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac{1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac{35 b e^2}{4 (b d-a e)^4 \sqrt{d+e x}}+\frac{\left (35 b^2 e^2\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 (b d-a e)^4}\\ &=\frac{35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac{1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac{35 b e^2}{4 (b d-a e)^4 \sqrt{d+e x}}+\frac{\left (35 b^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 (b d-a e)^4}\\ &=\frac{35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac{1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac{35 b e^2}{4 (b d-a e)^4 \sqrt{d+e x}}-\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0148984, size = 52, normalized size = 0.31 \[ -\frac{2 e^2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 206, normalized size = 1.2 \begin{align*} -{\frac{2\,{e}^{2}}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+{\frac{11\,{b}^{3}{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,a{b}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{13\,{b}^{3}d{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{b}^{2}{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \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 = 1.16703, size = 2472, normalized size = 14.8 \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 [B] time = 1.17609, size = 398, normalized size = 2.38 \begin{align*} \frac{35 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (9 \,{\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} + \frac{11 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{2} - 13 \, \sqrt{x e + d} b^{3} d e^{2} + 13 \, \sqrt{x e + d} a b^{2} e^{3}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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