Optimal. Leaf size=212 \[ -\frac{5 b e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{5 e (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}+\frac{5 b^{3/2} e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]
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Rubi [A] time = 0.110092, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {770, 21, 51, 63, 208} \[ -\frac{5 b e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{5 e (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}+\frac{5 b^{3/2} e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
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 )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{a+b x}{\left (a b+b^2 x\right )^3 (d+e x)^{5/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{(b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 e \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 b (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{(b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 e \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{(b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b e (a+b x)}{(b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 b e \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{(b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b e (a+b x)}{(b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 b \left (a b+b^2 x\right )\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 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{(b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b e (a+b x)}{(b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b^{3/2} e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0263483, size = 66, normalized size = 0.31 \[ -\frac{2 e (a+b x) \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 \sqrt{(a+b x)^2} (d+e x)^{3/2} (a e-b d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 242, normalized size = 1.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{2}}{3\, \left ( ae-bd \right ) ^{3}} \left ( 15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \left ( ex+d \right ) ^{3/2}x{b}^{3}e+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \left ( ex+d \right ) ^{3/2}a{b}^{2}e+15\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{b}^{2}{e}^{2}+10\,\sqrt{ \left ( ae-bd \right ) b}xab{e}^{2}+20\,\sqrt{ \left ( ae-bd \right ) b}x{b}^{2}de-2\,\sqrt{ \left ( ae-bd \right ) b}{a}^{2}{e}^{2}+14\,\sqrt{ \left ( ae-bd \right ) b}abde+3\,\sqrt{ \left ( ae-bd \right ) b}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.0975, size = 1592, normalized size = 7.51 \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(-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 = 1.24274, size = 667, normalized size = 3.15 \begin{align*} -\frac{5 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{{\left (b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} e^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{\sqrt{x e + d} b^{2} e^{2}}{{\left (b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} e^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \,{\left (b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} e^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\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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