Optimal. Leaf size=204 \[ \frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)}-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^4 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0677803, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)}-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^4 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}} \]
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)^{5/2}} \, 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)^{5/2}} \, 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^3 (b d-a e)^3}{e^3 (d+e x)^{5/2}}+\frac{3 b^4 (b d-a e)^2}{e^3 (d+e x)^{3/2}}-\frac{3 b^5 (b d-a e)}{e^3 \sqrt{d+e x}}+\frac{b^6 \sqrt{d+e x}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}-\frac{6 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) \sqrt{d+e x}}-\frac{6 b^2 (b d-a e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac{2 b^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0629554, size = 119, normalized size = 0.58 \[ -\frac{2 \sqrt{(a+b x)^2} \left (3 a^2 b e^2 (2 d+3 e x)+a^3 e^3-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (a+b x) (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.153, size = 131, normalized size = 0.6 \begin{align*} -{\frac{-2\,{x}^{3}{b}^{3}{e}^{3}-18\,{x}^{2}a{b}^{2}{e}^{3}+12\,{x}^{2}{b}^{3}d{e}^{2}+18\,x{a}^{2}b{e}^{3}-72\,xa{b}^{2}d{e}^{2}+48\,x{b}^{3}{d}^{2}e+2\,{a}^{3}{e}^{3}+12\,d{e}^{2}{a}^{2}b-48\,a{b}^{2}{d}^{2}e+32\,{b}^{3}{d}^{3}}{3\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10806, size = 169, normalized size = 0.83 \begin{align*} \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )}}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51978, size = 281, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19761, size = 273, normalized size = 1.34 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{8} \mathrm{sgn}\left (b x + a\right ) - 9 \, \sqrt{x e + d} b^{3} d e^{8} \mathrm{sgn}\left (b x + a\right ) + 9 \, \sqrt{x e + d} a b^{2} e^{9} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} b^{3} d^{2} \mathrm{sgn}\left (b x + a\right ) - b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) - 18 \,{\left (x e + d\right )} a b^{2} d e \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 9 \,{\left (x e + d\right )} a^{2} b e^{2} \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 )\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]