Optimal. Leaf size=209 \[ \frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{64 x}{5005 d^7 e^3 \sqrt{d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.314612, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1639, 793, 659, 192, 191} \[ \frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{64 x}{5005 d^7 e^3 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1639
Rule 793
Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{x^3}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{2 d^3 e^2-3 d^2 e^3 x-12 d e^4 x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 e^5}\\ &=-\frac{3 d}{14 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{-20 d^3 e^6+36 d^2 e^7 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{56 e^9}\\ &=\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{3 d}{14 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\left (9 d^2\right ) \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{182 e^3}\\ &=\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(36 d) \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1001 e^3}\\ &=\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^3}\\ &=\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{24 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1001 d e^3}\\ &=-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{96 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5005 d^3 e^3}\\ &=-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{64 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5005 d^5 e^3}\\ &=-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{64 x}{5005 d^7 e^3 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.198111, size = 137, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (315 d^7 e^2 x^2-540 d^6 e^3 x^3+160 d^5 e^4 x^4+776 d^4 e^5 x^5+384 d^3 e^6 x^6-224 d^2 e^7 x^7+360 d^8 e x+90 d^9-256 d e^8 x^8-64 e^9 x^9\right )}{5005 d^7 e^4 (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 132, normalized size = 0.6 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( -64\,{e}^{9}{x}^{9}-256\,{e}^{8}{x}^{8}d-224\,{e}^{7}{x}^{7}{d}^{2}+384\,{e}^{6}{x}^{6}{d}^{3}+776\,{e}^{5}{x}^{5}{d}^{4}+160\,{x}^{4}{d}^{5}{e}^{4}-540\,{x}^{3}{d}^{6}{e}^{3}+315\,{x}^{2}{d}^{7}{e}^{2}+360\,{d}^{8}xe+90\,{d}^{9} \right ) }{5005\,{e}^{4}{d}^{7} \left ( ex+d \right ) ^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \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 = 4.57104, size = 703, normalized size = 3.36 \begin{align*} \frac{90 \, e^{10} x^{10} + 360 \, d e^{9} x^{9} + 270 \, d^{2} e^{8} x^{8} - 720 \, d^{3} e^{7} x^{7} - 1260 \, d^{4} e^{6} x^{6} + 1260 \, d^{6} e^{4} x^{4} + 720 \, d^{7} e^{3} x^{3} - 270 \, d^{8} e^{2} x^{2} - 360 \, d^{9} e x - 90 \, d^{10} +{\left (64 \, e^{9} x^{9} + 256 \, d e^{8} x^{8} + 224 \, d^{2} e^{7} x^{7} - 384 \, d^{3} e^{6} x^{6} - 776 \, d^{4} e^{5} x^{5} - 160 \, d^{5} e^{4} x^{4} + 540 \, d^{6} e^{3} x^{3} - 315 \, d^{7} e^{2} x^{2} - 360 \, d^{8} e x - 90 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5005 \,{\left (d^{7} e^{14} x^{10} + 4 \, d^{8} e^{13} x^{9} + 3 \, d^{9} e^{12} x^{8} - 8 \, d^{10} e^{11} x^{7} - 14 \, d^{11} e^{10} x^{6} + 14 \, d^{13} e^{8} x^{4} + 8 \, d^{14} e^{7} x^{3} - 3 \, d^{15} e^{6} x^{2} - 4 \, d^{16} e^{5} x - d^{17} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]