Optimal. Leaf size=295 \[ \frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{2 e^6 (a+b x)}+\frac{10 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^6 (a+b x) (d+e x)^2}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^6 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.189606, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{2 e^6 (a+b x)}+\frac{10 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^6 (a+b x) (d+e x)^2}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^6 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^3} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{10 b^8 (b d-a e)^2}{e^5}-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^3}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^2}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)}-\frac{5 b^9 (b d-a e) (d+e x)}{e^5}+\frac{b^{10} (d+e x)^2}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{10 b^3 (b d-a e)^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x) (d+e x)^2}-\frac{5 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}-\frac{5 b^4 (b d-a e) (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x)}+\frac{b^5 (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}-\frac{10 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.113645, size = 248, normalized size = 0.84 \[ \frac{\sqrt{(a+b x)^2} \left (30 a^2 b^3 e^2 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+30 a^3 b^2 d e^3 (3 d+4 e x)-15 a^4 b e^4 (d+2 e x)-3 a^5 e^5+15 a b^4 e \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )-60 b^2 (d+e x)^2 (b d-a e)^3 \log (d+e x)+b^5 \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.203, size = 502, normalized size = 1.7 \begin{align*}{\frac{-360\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+360\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}+120\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}d{e}^{4}-180\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{4}+180\,\ln \left ( ex+d \right ){x}^{2}a{b}^{4}{d}^{2}{e}^{3}-60\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{2}{e}^{5}-3\,{a}^{5}{e}^{5}-27\,{b}^{5}{d}^{5}-15\,d{e}^{4}{a}^{4}b+6\,x{b}^{5}{d}^{4}e+15\,{x}^{4}a{b}^{4}{e}^{5}-5\,{x}^{4}{b}^{5}d{e}^{4}+60\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+20\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+63\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-30\,x{a}^{4}b{e}^{5}-120\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e-165\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+120\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-60\,{x}^{3}a{b}^{4}d{e}^{4}-150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+90\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}+2\,{x}^{5}{b}^{5}{e}^{5}+30\,xa{b}^{4}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{2}{e}^{3}-180\,\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}{e}^{2}+180\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e+120\,x{a}^{3}{b}^{2}d{e}^{4}-120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+105\,a{b}^{4}{d}^{4}e}{6\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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 = 1.62572, size = 840, normalized size = 2.85 \begin{align*} \frac{2 \, b^{5} e^{5} x^{5} - 27 \, b^{5} d^{5} + 105 \, a b^{4} d^{4} e - 150 \, a^{2} b^{3} d^{3} e^{2} + 90 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} - 3 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 3 \,{\left (21 \, b^{5} d^{3} e^{2} - 55 \, a b^{4} d^{2} e^{3} + 40 \, a^{2} b^{3} d e^{4}\right )} x^{2} + 6 \,{\left (b^{5} d^{4} e + 5 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} d^{5} - 3 \, a b^{4} d^{4} e + 3 \, a^{2} b^{3} d^{3} e^{2} - a^{3} b^{2} d^{2} e^{3} +{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 2 \,{\left (b^{5} d^{4} e - 3 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3} - a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\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 [A] time = 1.14085, size = 508, normalized size = 1.72 \begin{align*} -10 \,{\left (b^{5} d^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{4} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d e^{2} \mathrm{sgn}\left (b x + a\right ) - a^{3} b^{2} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, b^{5} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) - 9 \, b^{5} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 36 \, b^{5} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) - 90 \, a b^{4} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 60 \, a^{2} b^{3} x e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-9\right )} - \frac{{\left (9 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 35 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 50 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 30 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 10 \,{\left (b^{5} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{4} b e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]