Optimal. Leaf size=48 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0194524, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 37} \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, 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)^7} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}\\ \end{align*}
Mathematica [B] time = 0.0784895, size = 218, normalized size = 4.54 \[ -\frac{\sqrt{(a+b x)^2} \left (a^2 b^3 e^2 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^4 b e^4 (d+6 e x)+a^5 e^5+a b^4 e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.155, size = 283, normalized size = 5.9 \begin{align*} -{\frac{6\,{x}^{5}{b}^{5}{e}^{5}+15\,{x}^{4}a{b}^{4}{e}^{5}+15\,{x}^{4}{b}^{5}d{e}^{4}+20\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+20\,{x}^{3}a{b}^{4}d{e}^{4}+20\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+15\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+15\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+15\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+15\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+6\,x{a}^{4}b{e}^{5}+6\,x{a}^{3}{b}^{2}d{e}^{4}+6\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+6\,xa{b}^{4}{d}^{3}{e}^{2}+6\,x{b}^{5}{d}^{4}e+{a}^{5}{e}^{5}+d{e}^{4}{a}^{4}b+{a}^{3}{b}^{2}{d}^{2}{e}^{3}+{a}^{2}{b}^{3}{d}^{3}{e}^{2}+a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{6\, \left ( ex+d \right ) ^{6}{e}^{6} \left ( bx+a \right ) ^{5}} \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 [B] time = 1.64554, size = 595, normalized size = 12.4 \begin{align*} -\frac{6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \,{\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \,{\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} 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 [B] time = 1.21132, size = 508, normalized size = 10.58 \begin{align*} -\frac{{\left (6 \, b^{5} x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{5} d x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{5} d^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, b^{5} d^{4} x e \mathrm{sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 20 \, a b^{4} d x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{4} d^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{3} x e^{2} \mathrm{sgn}\left (b x + a\right ) + a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} d x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{4} b x e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]