Optimal. Leaf size=144 \[ \frac{5 e \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^3 d^3}-\frac{5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2} \]
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Rubi [A] time = 0.0953684, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {626, 47, 50, 63, 208} \[ \frac{5 e \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^3 d^3}-\frac{5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{(d+e x)^{5/2}}{(a e+c d x)^2} \, dx\\ &=-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{(5 e) \int \frac{(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{\left (5 e \left (c d^2-a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac{5 e \left (c d^2-a e^2\right ) \sqrt{d+e x}}{c^3 d^3}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{\left (5 e \left (c d^2-a e^2\right )^2\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{2 c^3 d^3}\\ &=\frac{5 e \left (c d^2-a e^2\right ) \sqrt{d+e x}}{c^3 d^3}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{\left (5 \left (c d^2-a e^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c d^2}{e}+a e+\frac{c d x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{c^3 d^3}\\ &=\frac{5 e \left (c d^2-a e^2\right ) \sqrt{d+e x}}{c^3 d^3}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}-\frac{5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0175845, size = 59, normalized size = 0.41 \[ \frac{2 e (d+e x)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{7 \left (a e^2-c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.204, size = 314, normalized size = 2.2 \begin{align*}{\frac{2\,e}{3\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}{d}^{3}}}+4\,{\frac{e\sqrt{ex+d}}{{c}^{2}d}}-{\frac{{a}^{2}{e}^{5}}{{c}^{3}{d}^{3} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+2\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{2}d \left ( cdex+a{e}^{2} \right ) }}-{\frac{de}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+5\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}{d}^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-10\,{\frac{a{e}^{3}}{{c}^{2}d\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+5\,{\frac{de}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \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 = 1.97144, size = 855, normalized size = 5.94 \begin{align*} \left [\frac{15 \,{\left (a c d^{2} e^{2} - a^{2} e^{4} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} + 2 \,{\left (7 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}, -\frac{15 \,{\left (a c d^{2} e^{2} - a^{2} e^{4} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac{\sqrt{e x + d} c d \sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) -{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} + 2 \,{\left (7 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\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 [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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