Optimal. Leaf size=157 \[ \frac{2 e \sqrt{d+e x} \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{c^3}+\frac{2 e (d+e x)^{3/2} (2 c d-b e)}{3 c^2}+\frac{2 (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}-\frac{2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e (d+e x)^{5/2}}{5 c} \]
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Rubi [A] time = 0.387282, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {703, 824, 826, 1166, 208} \[ \frac{2 e \sqrt{d+e x} \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{c^3}+\frac{2 e (d+e x)^{3/2} (2 c d-b e)}{3 c^2}+\frac{2 (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}-\frac{2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e (d+e x)^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 703
Rule 824
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{b x+c x^2} \, dx &=\frac{2 e (d+e x)^{5/2}}{5 c}+\frac{\int \frac{(d+e x)^{3/2} \left (c d^2+e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{c}\\ &=\frac{2 e (2 c d-b e) (d+e x)^{3/2}}{3 c^2}+\frac{2 e (d+e x)^{5/2}}{5 c}+\frac{\int \frac{\sqrt{d+e x} \left (c^2 d^3+e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{c^2}\\ &=\frac{2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{c^3}+\frac{2 e (2 c d-b e) (d+e x)^{3/2}}{3 c^2}+\frac{2 e (d+e x)^{5/2}}{5 c}+\frac{\int \frac{c^3 d^4+e (2 c d-b e) \left (2 c^2 d^2-2 b c d e+b^2 e^2\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{c^3}\\ &=\frac{2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{c^3}+\frac{2 e (2 c d-b e) (d+e x)^{3/2}}{3 c^2}+\frac{2 e (d+e x)^{5/2}}{5 c}+\frac{2 \operatorname{Subst}\left (\int \frac{c^3 d^4 e-d e (2 c d-b e) \left (2 c^2 d^2-2 b c d e+b^2 e^2\right )+e (2 c d-b e) \left (2 c^2 d^2-2 b c d e+b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{c^3}\\ &=\frac{2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{c^3}+\frac{2 e (2 c d-b e) (d+e x)^{3/2}}{3 c^2}+\frac{2 e (d+e x)^{5/2}}{5 c}+\frac{\left (2 c d^4\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}-\frac{\left (2 (c d-b e)^4\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b c^3}\\ &=\frac{2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{c^3}+\frac{2 e (2 c d-b e) (d+e x)^{3/2}}{3 c^2}+\frac{2 e (d+e x)^{5/2}}{5 c}-\frac{2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.206247, size = 138, normalized size = 0.88 \[ \frac{2 e \sqrt{d+e x} \left (15 b^2 e^2-5 b c e (10 d+e x)+c^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )}{15 c^3}+\frac{2 (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}-\frac{2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.246, size = 336, normalized size = 2.1 \begin{align*}{\frac{2\,e}{5\,c} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{2\,b{e}^{2}}{3\,{c}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{4\,de}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{e}^{3}{b}^{2}\sqrt{ex+d}}{{c}^{3}}}-6\,{\frac{bd{e}^{2}\sqrt{ex+d}}{{c}^{2}}}+6\,{\frac{e{d}^{2}\sqrt{ex+d}}{c}}-2\,{\frac{{b}^{3}{e}^{4}}{{c}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+8\,{\frac{{e}^{3}{b}^{2}d}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{b{e}^{2}{d}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+8\,{\frac{e{d}^{3}}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{c{d}^{4}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{d}^{7/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \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 [A] time = 9.5173, size = 1820, normalized size = 11.59 \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 [A] time = 92.468, size = 162, normalized size = 1.03 \begin{align*} \frac{2 e \left (d + e x\right )^{\frac{5}{2}}}{5 c} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- 2 b e^{2} + 4 c d e\right )}{3 c^{2}} + \frac{\sqrt{d + e x} \left (2 b^{2} e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e\right )}{c^{3}} + \frac{2 d^{4} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b \sqrt{- d}} - \frac{2 \left (b e - c d\right )^{4} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{b c^{4} \sqrt{\frac{b e - c d}{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25215, size = 309, normalized size = 1.97 \begin{align*} \frac{2 \, d^{4} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} - \frac{2 \,{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{4} e + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d e + 45 \, \sqrt{x e + d} c^{4} d^{2} e - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} e^{2} - 45 \, \sqrt{x e + d} b c^{3} d e^{2} + 15 \, \sqrt{x e + d} b^{2} c^{2} e^{3}\right )}}{15 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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