Optimal. Leaf size=264 \[ -\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^5 (a+b x) (d+e x)^{9/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^5 (a+b x) (d+e x)^{11/2}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^{5/2}} \]
[Out]
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Rubi [A] time = 0.306986, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^5 (a+b x) (d+e x)^{9/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^5 (a+b x) (d+e x)^{11/2}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(13/2),x]
[Out]
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Rubi in Sympy [A] time = 35.367, size = 212, normalized size = 0.8 \[ - \frac{128 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{4} \left (d + e x\right )^{\frac{5}{2}}} + \frac{256 b^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3465 e^{5} \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}} - \frac{32 b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{3} \left (d + e x\right )^{\frac{7}{2}}} - \frac{16 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 e^{2} \left (d + e x\right )^{\frac{9}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{11 e \left (d + e x\right )^{\frac{11}{2}}} \]
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)**(3/2)/(e*x+d)**(13/2),x)
[Out]
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Mathematica [A] time = 0.289117, size = 119, normalized size = 0.45 \[ \frac{2 \sqrt{(a+b x)^2} \left (2772 b^3 (d+e x)^3 (b d-a e)-2970 b^2 (d+e x)^2 (b d-a e)^2+1540 b (d+e x) (b d-a e)^3-315 (b d-a e)^4-1155 b^4 (d+e x)^4\right )}{3465 e^5 (a+b x) (d+e x)^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(13/2),x]
[Out]
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Maple [A] time = 0.013, size = 202, normalized size = 0.8 \[ -{\frac{2310\,{x}^{4}{b}^{4}{e}^{4}+5544\,{x}^{3}a{b}^{3}{e}^{4}+3696\,{x}^{3}{b}^{4}d{e}^{3}+5940\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+4752\,{x}^{2}a{b}^{3}d{e}^{3}+3168\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+3080\,x{a}^{3}b{e}^{4}+2640\,x{a}^{2}{b}^{2}d{e}^{3}+2112\,xa{b}^{3}{d}^{2}{e}^{2}+1408\,x{b}^{4}{d}^{3}e+630\,{a}^{4}{e}^{4}+560\,{a}^{3}bd{e}^{3}+480\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+384\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{3465\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \]
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)^(3/2)/(e*x+d)^(13/2),x)
[Out]
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Maxima [A] time = 0.737982, size = 531, normalized size = 2.01 \[ -\frac{2 \,{\left (231 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 40 \, a b^{2} d^{2} e + 70 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 99 \,{\left (2 \, b^{3} d e^{2} + 5 \, a b^{2} e^{3}\right )} x^{2} + 11 \,{\left (8 \, b^{3} d^{2} e + 20 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} a}{1155 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )} \sqrt{e x + d}} - \frac{2 \,{\left (1155 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} + 144 \, a b^{2} d^{3} e + 120 \, a^{2} b d^{2} e^{2} + 70 \, a^{3} d e^{3} + 231 \,{\left (8 \, b^{3} d e^{3} + 9 \, a b^{2} e^{4}\right )} x^{3} + 99 \,{\left (16 \, b^{3} d^{2} e^{2} + 18 \, a b^{2} d e^{3} + 15 \, a^{2} b e^{4}\right )} x^{2} + 11 \,{\left (64 \, b^{3} d^{3} e + 72 \, a b^{2} d^{2} e^{2} + 60 \, a^{2} b d e^{3} + 35 \, a^{3} e^{4}\right )} x\right )} b}{3465 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(13/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282472, size = 319, normalized size = 1.21 \[ -\frac{2 \,{\left (1155 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 192 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 924 \,{\left (2 \, b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 198 \,{\left (8 \, b^{4} d^{2} e^{2} + 12 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 44 \,{\left (16 \, b^{4} d^{3} e + 24 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x\right )}}{3465 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(13/2),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)**(3/2)/(e*x+d)**(13/2),x)
[Out]
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GIAC/XCAS [A] time = 0.322337, size = 414, normalized size = 1.57 \[ -\frac{2 \,{\left (1155 \,{\left (x e + d\right )}^{4} b^{4}{\rm sign}\left (b x + a\right ) - 2772 \,{\left (x e + d\right )}^{3} b^{4} d{\rm sign}\left (b x + a\right ) + 2970 \,{\left (x e + d\right )}^{2} b^{4} d^{2}{\rm sign}\left (b x + a\right ) - 1540 \,{\left (x e + d\right )} b^{4} d^{3}{\rm sign}\left (b x + a\right ) + 315 \, b^{4} d^{4}{\rm sign}\left (b x + a\right ) + 2772 \,{\left (x e + d\right )}^{3} a b^{3} e{\rm sign}\left (b x + a\right ) - 5940 \,{\left (x e + d\right )}^{2} a b^{3} d e{\rm sign}\left (b x + a\right ) + 4620 \,{\left (x e + d\right )} a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) - 1260 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 2970 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4620 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) + 1890 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 1540 \,{\left (x e + d\right )} a^{3} b e^{3}{\rm sign}\left (b x + a\right ) - 1260 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + 315 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{3465 \,{\left (x e + d\right )}^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(13/2),x, algorithm="giac")
[Out]