Optimal. Leaf size=362 \[ -\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^7 (a+b x) (d+e x)^7}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{8 e^7 (a+b x) (d+e x)^8}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^7 (a+b x) (d+e x)^9} \]
[Out]
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Rubi [A] time = 0.60142, 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 \[ -\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^7 (a+b x) (d+e x)^7}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{8 e^7 (a+b x) (d+e x)^8}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^7 (a+b x) (d+e x)^9} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^13,x]
[Out]
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Rubi in Sympy [A] time = 89.4969, size = 238, normalized size = 0.66 \[ \frac{b^{5} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{5544 \left (d + e x\right )^{7} \left (a e - b d\right )^{6}} - \frac{b^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{792 \left (d + e x\right )^{8} \left (a e - b d\right )^{5}} + \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{198 \left (d + e x\right )^{9} \left (a e - b d\right )^{4}} - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{66 \left (d + e x\right )^{10} \left (a e - b d\right )^{3}} + \frac{5 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{132 \left (d + e x\right )^{11} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{12 \left (d + e x\right )^{12} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**13,x)
[Out]
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Mathematica [A] time = 0.231814, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+56 a^3 b^3 e^3 \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+21 a^2 b^4 e^2 \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+6 a b^5 e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )}{5544 e^7 (a+b x) (d+e x)^{12}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^13,x]
[Out]
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Maple [A] time = 0.017, size = 392, normalized size = 1.1 \[ -{\frac{924\,{x}^{6}{b}^{6}{e}^{6}+4752\,{x}^{5}a{b}^{5}{e}^{6}+792\,{x}^{5}{b}^{6}d{e}^{5}+10395\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+2970\,{x}^{4}a{b}^{5}d{e}^{5}+495\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+12320\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+4620\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+1320\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+220\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+8316\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+3696\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+1386\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+396\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+66\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+3024\,x{a}^{5}b{e}^{6}+1512\,x{a}^{4}{b}^{2}d{e}^{5}+672\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+252\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+72\,xa{b}^{5}{d}^{4}{e}^{2}+12\,x{b}^{6}{d}^{5}e+462\,{a}^{6}{e}^{6}+252\,{a}^{5}bd{e}^{5}+126\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+56\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+21\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+6\,{d}^{5}a{b}^{5}e+{b}^{6}{d}^{6}}{5544\,{e}^{7} \left ( ex+d \right ) ^{12} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^13,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^13,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282701, size = 640, normalized size = 1.77 \[ -\frac{924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \,{\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \,{\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \,{\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \,{\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \,{\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^13,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**13,x)
[Out]
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GIAC/XCAS [A] time = 0.294513, size = 702, normalized size = 1.94 \[ -\frac{{\left (924 \, b^{6} x^{6} e^{6}{\rm sign}\left (b x + a\right ) + 792 \, b^{6} d x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 495 \, b^{6} d^{2} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 220 \, b^{6} d^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 66 \, b^{6} d^{4} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 12 \, b^{6} d^{5} x e{\rm sign}\left (b x + a\right ) + b^{6} d^{6}{\rm sign}\left (b x + a\right ) + 4752 \, a b^{5} x^{5} e^{6}{\rm sign}\left (b x + a\right ) + 2970 \, a b^{5} d x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 1320 \, a b^{5} d^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 396 \, a b^{5} d^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 72 \, a b^{5} d^{4} x e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 10395 \, a^{2} b^{4} x^{4} e^{6}{\rm sign}\left (b x + a\right ) + 4620 \, a^{2} b^{4} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 1386 \, a^{2} b^{4} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 252 \, a^{2} b^{4} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 12320 \, a^{3} b^{3} x^{3} e^{6}{\rm sign}\left (b x + a\right ) + 3696 \, a^{3} b^{3} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 672 \, a^{3} b^{3} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 56 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 8316 \, a^{4} b^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) + 1512 \, a^{4} b^{2} d x e^{5}{\rm sign}\left (b x + a\right ) + 126 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 3024 \, a^{5} b x e^{6}{\rm sign}\left (b x + a\right ) + 252 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 462 \, a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{5544 \,{\left (x e + d\right )}^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^13,x, algorithm="giac")
[Out]