Optimal. Leaf size=210 \[ \frac{9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}+\frac{3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{9 e \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2} \]
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Rubi [A] time = 0.176113, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}+\frac{3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{9 e \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2} \]
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)^{13/2}}{\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)^{9/2}}{(a e+c d x)^2} \, dx\\ &=-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{(9 e) \int \frac{(d+e x)^{7/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )\right ) \int \frac{(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )^2\right ) \int \frac{(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^3 d^3}\\ &=\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )^3\right ) \int \frac{\sqrt{d+e x}}{a e+c d x} \, dx}{2 c^4 d^4}\\ &=\frac{9 e \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^5 d^5}+\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )^4\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{2 c^5 d^5}\\ &=\frac{9 e \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^5 d^5}+\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 \left (c d^2-a e^2\right )^4\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^5 d^5}\\ &=\frac{9 e \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^5 d^5}+\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0251819, size = 59, normalized size = 0.28 \[ \frac{2 e (d+e x)^{11/2} \, _2F_1\left (2,\frac{11}{2};\frac{13}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{11 \left (a e^2-c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.203, size = 628, normalized size = 3. \begin{align*}{\frac{2\,e}{7\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a{e}^{3}}{5\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{4\,e}{5\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+2\,{\frac{ \left ( ex+d \right ) ^{3/2}{a}^{2}{e}^{5}}{{c}^{4}{d}^{4}}}-4\,{\frac{ \left ( ex+d \right ) ^{3/2}a{e}^{3}}{{c}^{3}{d}^{2}}}+2\,{\frac{e \left ( ex+d \right ) ^{3/2}}{{c}^{2}}}-8\,{\frac{{a}^{3}{e}^{7}\sqrt{ex+d}}{{c}^{5}{d}^{5}}}+24\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{4}{d}^{3}}}-24\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}d}}+8\,{\frac{de\sqrt{ex+d}}{{c}^{2}}}-{\frac{{a}^{4}{e}^{9}}{{c}^{5}{d}^{5} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+4\,{\frac{{a}^{3}{e}^{7}\sqrt{ex+d}}{{c}^{4}{d}^{3} \left ( cdex+a{e}^{2} \right ) }}-6\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{3}d \left ( cdex+a{e}^{2} \right ) }}+4\,{\frac{d\sqrt{ex+d}a{e}^{3}}{{c}^{2} \left ( cdex+a{e}^{2} \right ) }}-{\frac{e{d}^{3}}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+9\,{\frac{{a}^{4}{e}^{9}}{{c}^{5}{d}^{5}\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 ) }-36\,{\frac{{a}^{3}{e}^{7}}{{c}^{4}{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 ) }+54\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}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 ) }-36\,{\frac{ad{e}^{3}}{{c}^{2}\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 ) }+9\,{\frac{e{d}^{3}}{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 [B] time = 2.03539, size = 1629, normalized size = 7.76 \begin{align*} \left [\frac{315 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\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 (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \,{\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt{e x + d}}{70 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac{315 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\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 (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \,{\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} 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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