Optimal. Leaf size=186 \[ -\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0856378, antiderivative size = 186, 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} \[ -\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]
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 )^{3/2}}{(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 )^3}{(d+e x)^3} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{b^6}{e^3}-\frac{b^3 (b d-a e)^3}{e^3 (d+e x)^3}+\frac{3 b^4 (b d-a e)^2}{e^3 (d+e x)^2}-\frac{3 b^5 (b d-a e)}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac{(b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^2}-\frac{3 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) (d+e x)}-\frac{3 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0680947, size = 131, normalized size = 0.7 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^2 b e^2 (d+2 e x)+a^3 e^3-3 a b^2 d e (3 d+4 e x)+6 b^2 (d+e x)^2 (b d-a e) \log (d+e x)+b^3 \left (4 d^2 e x+5 d^3-4 d e^2 x^2-2 e^3 x^3\right )\right )}{2 e^4 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.202, size = 219, normalized size = 1.2 \begin{align*}{\frac{6\,\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{3}-6\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}+2\,{x}^{3}{b}^{3}{e}^{3}+12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}-12\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e+4\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}-6\,x{a}^{2}b{e}^{3}+12\,xa{b}^{2}d{e}^{2}-4\,x{b}^{3}{d}^{2}e-{a}^{3}{e}^{3}-3\,d{e}^{2}{a}^{2}b+9\,a{b}^{2}{d}^{2}e-5\,{b}^{3}{d}^{3}}{2\, \left ( bx+a \right ) ^{3}{e}^{4} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 1.50743, size = 375, normalized size = 2.02 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d e^{2} x^{2} - 5 \, b^{3} d^{3} + 9 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 2 \,{\left (2 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} d^{3} - a b^{2} d^{2} e +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (b^{3} d^{2} e - a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\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 = 1.21325, size = 230, normalized size = 1.24 \begin{align*} b^{3} x e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) - 3 \,{\left (b^{3} d \mathrm{sgn}\left (b x + a\right ) - a b^{2} e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) - 9 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) + a^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \,{\left (b^{3} d^{2} e \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{2} d e^{2} \mathrm{sgn}\left (b x + a\right ) + a^{2} b e^{3} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]