Optimal. Leaf size=359 \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.198234, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^{12}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{12}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{11}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{10}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^9}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^8}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^7}+\frac{b^6}{e^6 (d+e x)^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{3 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{5 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac{5 b^3 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^8}-\frac{15 b^4 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac{b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^6}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.116632, size = 295, normalized size = 0.82 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 b^4 e^2 \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+35 a^3 b^3 e^3 \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+126 a^5 b e^5 (d+11 e x)+210 a^6 e^6+5 a b^5 e \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )\right )}{2310 e^7 (a+b x) (d+e x)^{11}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 392, normalized size = 1.1 \begin{align*} -{\frac{462\,{x}^{6}{b}^{6}{e}^{6}+2310\,{x}^{5}a{b}^{5}{e}^{6}+462\,{x}^{5}{b}^{6}d{e}^{5}+4950\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+1650\,{x}^{4}a{b}^{5}d{e}^{5}+330\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+5775\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+2475\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+825\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+165\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+3850\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+1925\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+825\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+275\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+55\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1386\,x{a}^{5}b{e}^{6}+770\,x{a}^{4}{b}^{2}d{e}^{5}+385\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+165\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+55\,xa{b}^{5}{d}^{4}{e}^{2}+11\,x{b}^{6}{d}^{5}e+210\,{a}^{6}{e}^{6}+126\,d{e}^{5}{a}^{5}b+70\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+35\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+15\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+5\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{2310\,{e}^{7} \left ( ex+d \right ) ^{11} \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 [A] time = 1.55396, size = 994, normalized size = 2.77 \begin{align*} -\frac{462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \,{\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \,{\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \,{\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \,{\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \,{\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\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 = 1.17277, size = 702, normalized size = 1.96 \begin{align*} -\frac{{\left (462 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 462 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 330 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 165 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 55 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 11 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 2310 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 1650 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 825 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 275 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 55 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 4950 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 2475 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 165 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5775 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 1925 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 385 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 3850 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 770 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 70 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 1386 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 126 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 210 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{2310 \,{\left (x e + d\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]