Optimal. Leaf size=48 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]
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Rubi [A] time = 0.0708958, 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 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 8.97975, size = 44, normalized size = 0.92 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 \left (d + e x\right )^{6} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**7,x)
[Out]
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Mathematica [B] time = 0.149723, size = 218, normalized size = 4.54 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5 e^5+a^4 b e^4 (d+6 e x)+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^2 b^3 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+a b^4 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]
[Out]
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Maple [B] time = 0.011, size = 283, normalized size = 5.9 \[ -{\frac{6\,{x}^{5}{b}^{5}{e}^{5}+15\,{x}^{4}a{b}^{4}{e}^{5}+15\,{x}^{4}{b}^{5}d{e}^{4}+20\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+20\,{x}^{3}a{b}^{4}d{e}^{4}+20\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+15\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+15\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+15\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+15\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+6\,x{a}^{4}b{e}^{5}+6\,x{a}^{3}{b}^{2}d{e}^{4}+6\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+6\,xa{b}^{4}{d}^{3}{e}^{2}+6\,x{b}^{5}{d}^{4}e+{a}^{5}{e}^{5}+{a}^{4}bd{e}^{4}+{a}^{3}{b}^{2}{d}^{2}{e}^{3}+{a}^{2}{b}^{3}{d}^{3}{e}^{2}+a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{6\, \left ( ex+d \right ) ^{6}{e}^{6} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,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)/(e*x + d)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210302, size = 405, normalized size = 8.44 \[ -\frac{6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \,{\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \,{\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^7,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**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.222258, size = 508, normalized size = 10.58 \[ -\frac{{\left (6 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 15 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 20 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 15 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 20 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 15 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^7,x, algorithm="giac")
[Out]