Optimal. Leaf size=152 \[ -\frac{15 e^2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{7/2} d^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 c^3 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.100075, antiderivative size = 152, 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} \[ -\frac{15 e^2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{7/2} d^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 c^3 d^3} \]
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)^{11/2}}{\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)^{5/2}}{(a e+c d x)^3} \, dx\\ &=-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{(5 e) \int \frac{(d+e x)^{3/2}}{(a e+c d x)^2} \, dx}{4 c d}\\ &=-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{\left (15 e^2\right ) \int \frac{\sqrt{d+e x}}{a e+c d x} \, dx}{8 c^2 d^2}\\ &=\frac{15 e^2 \sqrt{d+e x}}{4 c^3 d^3}-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{\left (15 e^2 \left (c d^2-a e^2\right )\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{8 c^3 d^3}\\ &=\frac{15 e^2 \sqrt{d+e x}}{4 c^3 d^3}-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{\left (15 e \left (c d^2-a e^2\right )\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^3 d^3}\\ &=\frac{15 e^2 \sqrt{d+e x}}{4 c^3 d^3}-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}-\frac{15 e^2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{7/2} d^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0181947, size = 61, normalized size = 0.4 \[ \frac{2 e^2 (d+e x)^{7/2} \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{7 \left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.202, size = 288, normalized size = 1.9 \begin{align*} 2\,{\frac{{e}^{2}\sqrt{ex+d}}{{c}^{3}{d}^{3}}}+{\frac{9\,{e}^{4}a}{4\,{c}^{2}{d}^{2} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{e}^{2}}{4\,c \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{e}^{6}{a}^{2}}{4\,{c}^{3}{d}^{3} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{7\,{e}^{4}a}{2\,{c}^{2}d \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{7\,d{e}^{2}}{4\,c \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,{e}^{4}a}{4\,{c}^{3}{d}^{3}}\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{15\,{e}^{2}}{4\,{c}^{2}d}\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 [A] time = 2.00768, size = 892, normalized size = 5.87 \begin{align*} \left [\frac{15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\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^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 5 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} -{\left (9 \, c^{2} d^{3} e - 25 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}, -\frac{15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\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^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 5 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} -{\left (9 \, c^{2} d^{3} e - 25 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} 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]