Optimal. Leaf size=362 \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^7 (a+b x) (d+e x)^9}-\frac{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{14 e^7 (a+b x) (d+e x)^{14}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.197929, antiderivative size = 362, 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} \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^7 (a+b x) (d+e x)^9}-\frac{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{14 e^7 (a+b x) (d+e x)^{14}} \]
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)^{15}} \, 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)^{15}} \, 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)^{15}} \, 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)^{15}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{14}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{13}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{12}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^{11}}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^{10}}+\frac{b^6}{e^6 (d+e x)^9}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac{6 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{5 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{20 b^3 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{3 b^4 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{2 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}\\ \end{align*}
Mathematica [A] time = 0.116926, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (36 a^2 b^4 e^2 \left (91 d^2 e^2 x^2+14 d^3 e x+d^4+364 d e^3 x^3+1001 e^4 x^4\right )+120 a^3 b^3 e^3 \left (14 d^2 e x+d^3+91 d e^2 x^2+364 e^3 x^3\right )+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+792 a^5 b e^5 (d+14 e x)+1716 a^6 e^6+8 a b^5 e \left (91 d^3 e^2 x^2+364 d^2 e^3 x^3+14 d^4 e x+d^5+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+14 d^5 e x+d^6+2002 d e^5 x^5+3003 e^6 x^6\right )\right )}{24024 e^7 (a+b x) (d+e x)^{14}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 392, normalized size = 1.1 \begin{align*} -{\frac{3003\,{x}^{6}{b}^{6}{e}^{6}+16016\,{x}^{5}a{b}^{5}{e}^{6}+2002\,{x}^{5}{b}^{6}d{e}^{5}+36036\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+8008\,{x}^{4}a{b}^{5}d{e}^{5}+1001\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+43680\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+13104\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+2912\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+364\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+30030\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+10920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+3276\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+728\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+91\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+11088\,x{a}^{5}b{e}^{6}+4620\,x{a}^{4}{b}^{2}d{e}^{5}+1680\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+504\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+112\,xa{b}^{5}{d}^{4}{e}^{2}+14\,x{b}^{6}{d}^{5}e+1716\,{a}^{6}{e}^{6}+792\,d{e}^{5}{a}^{5}b+330\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+120\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+36\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+8\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{24024\,{e}^{7} \left ( ex+d \right ) ^{14} \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 = 1.56507, size = 1104, normalized size = 3.05 \begin{align*} -\frac{3003 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \,{\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \,{\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \,{\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \,{\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \,{\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \,{\left (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} 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 = 1.16863, size = 702, normalized size = 1.94 \begin{align*} -\frac{{\left (3003 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 2002 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 1001 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 364 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 91 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 14 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 16016 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 8008 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 2912 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 728 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 112 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 36036 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 13104 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 3276 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 504 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 36 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 43680 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 10920 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 1680 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 120 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 30030 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 4620 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 330 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 11088 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 792 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 1716 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{24024 \,{\left (x e + d\right )}^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]