Optimal. Leaf size=186 \[ \frac{35 e^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac{35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.137784, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{35 e^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac{35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
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 )^3} \, dx &=\int \frac{(d+e x)^{7/2}}{(a e+c d x)^3} \, dx\\ &=-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{(7 e) \int \frac{(d+e x)^{5/2}}{(a e+c d x)^2} \, dx}{4 c d}\\ &=-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{\left (35 e^2\right ) \int \frac{(d+e x)^{3/2}}{a e+c d x} \, dx}{8 c^2 d^2}\\ &=\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{\left (35 e^2 \left (c d^2-a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{a e+c d x} \, dx}{8 c^3 d^3}\\ &=\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{4 c^4 d^4}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{8 c^4 d^4}\\ &=\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{4 c^4 d^4}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{\left (35 e \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 )}{4 c^4 d^4}\\ &=\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{4 c^4 d^4}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}-\frac{35 e^2 \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 )}{4 c^{9/2} d^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0199576, size = 61, normalized size = 0.33 \[ \frac{2 e^2 (d+e x)^{9/2} \, _2F_1\left (3,\frac{9}{2};\frac{11}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{9 \left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.204, size = 449, normalized size = 2.4 \begin{align*}{\frac{2\,{e}^{2}}{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-6\,{\frac{{e}^{4}a\sqrt{ex+d}}{{c}^{4}{d}^{4}}}+6\,{\frac{{e}^{2}\sqrt{ex+d}}{{c}^{3}{d}^{2}}}-{\frac{13\,{e}^{6}{a}^{2}}{4\,{c}^{3}{d}^{3} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{e}^{4}a}{2\,{c}^{2}d \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{13\,d{e}^{2}}{4\,c \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{11\,{e}^{8}{a}^{3}}{4\,{c}^{4}{d}^{4} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{33\,{e}^{6}{a}^{2}}{4\,{c}^{3}{d}^{2} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{33\,{e}^{4}a}{4\,{c}^{2} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{11\,{d}^{2}{e}^{2}}{4\,c \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{e}^{6}{a}^{2}}{4\,{c}^{4}{d}^{4}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}}-{\frac{35\,{e}^{4}a}{2\,{c}^{3}{d}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}}+{\frac{35\,{e}^{2}}{4\,{c}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.03103, size = 1289, normalized size = 6.93 \begin{align*} \left [\frac{105 \,{\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\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 (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \,{\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} -{\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac{105 \,{\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\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 (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \,{\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} -{\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{12 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]