Optimal. Leaf size=202 \[ \frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^4 (a+b x)}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) \sqrt{d+e x}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^4 (a+b x) (d+e x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.066247, antiderivative size = 202, 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} \[ \frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^4 (a+b x)}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) \sqrt{d+e x}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^4 (a+b x) (d+e x)^{5/2}} \]
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)^{7/2}} \, 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)^{7/2}} \, 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)^{7/2}}+\frac{3 b^4 (b d-a e)^2}{e^3 (d+e x)^{5/2}}-\frac{3 b^5 (b d-a e)}{e^3 (d+e x)^{3/2}}+\frac{b^6}{e^3 \sqrt{d+e x}}\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}}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{2 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)^{3/2}}+\frac{6 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 b^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0647071, size = 118, normalized size = 0.58 \[ -\frac{2 \sqrt{(a+b x)^2} \left (a^2 b e^2 (2 d+5 e x)+a^3 e^3+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )+b^3 \left (-\left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{5 e^4 (a+b x) (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.153, size = 131, normalized size = 0.7 \begin{align*} -{\frac{-10\,{x}^{3}{b}^{3}{e}^{3}+30\,{x}^{2}a{b}^{2}{e}^{3}-60\,{x}^{2}{b}^{3}d{e}^{2}+10\,x{a}^{2}b{e}^{3}+40\,xa{b}^{2}d{e}^{2}-80\,x{b}^{3}{d}^{2}e+2\,{a}^{3}{e}^{3}+4\,d{e}^{2}{a}^{2}b+16\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{5\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.18217, size = 185, normalized size = 0.92 \begin{align*} \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )}}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64484, size = 298, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{5 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} 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{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21597, size = 265, normalized size = 1.31 \begin{align*} 2 \, \sqrt{x e + d} b^{3} e^{\left (-4\right )} \mathrm{sgn}\left (b x + a\right ) + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} b^{3} d \mathrm{sgn}\left (b x + a\right ) - 5 \,{\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 ) - 15 \,{\left (x e + d\right )}^{2} a b^{2} e \mathrm{sgn}\left (b x + a\right ) + 10 \,{\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 ) - 5 \,{\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 )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]