3.1553 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=48 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]

[Out]

((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(b*d - a*e)*(d + e*x)^4)

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Rubi [A]  time = 0.0701883, 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 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(b*d - a*e)*(d + e*x)^4)

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Rubi in Sympy [A]  time = 10.2688, size = 44, normalized size = 0.92 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 \left (d + e x\right )^{4} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**5,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*(d + e*x)**4*(a*e - b*d))

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Mathematica [B]  time = 0.0833553, size = 109, normalized size = 2.27 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{4 e^4 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x +
6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)))/(4*e^4*(a + b*x)*
(d + e*x)^4)

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Maple [B]  time = 0.011, size = 128, normalized size = 2.7 \[ -{\frac{4\,{x}^{3}{b}^{3}{e}^{3}+6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+4\,x{a}^{2}b{e}^{3}+4\,xa{b}^{2}d{e}^{2}+4\,x{b}^{3}{d}^{2}e+{a}^{3}{e}^{3}+{a}^{2}bd{e}^{2}+a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{4\, \left ( ex+d \right ) ^{4}{e}^{4} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^5,x)

[Out]

-1/4*(4*b^3*e^3*x^3+6*a*b^2*e^3*x^2+6*b^3*d*e^2*x^2+4*a^2*b*e^3*x+4*a*b^2*d*e^2*
x+4*b^3*d^2*e*x+a^3*e^3+a^2*b*d*e^2+a*b^2*d^2*e+b^3*d^3)*((b*x+a)^2)^(3/2)/(e*x+
d)^4/e^4/(b*x+a)^3

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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)^(3/2)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206709, size = 193, normalized size = 4.02 \[ -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^5,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*e^3*x^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2 + a^3*e^3 + 6*(b^3*d*e
^2 + a*b^2*e^3)*x^2 + 4*(b^3*d^2*e + a*b^2*d*e^2 + a^2*b*e^3)*x)/(e^8*x^4 + 4*d*
e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)

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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)**(3/2)/(e*x+d)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214259, size = 224, normalized size = 4.67 \[ -\frac{{\left (4 \, b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 6 \, a b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 4 \, a b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 4 \, a^{2} b x e^{3}{\rm sign}\left (b x + a\right ) + a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{4 \,{\left (x e + d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^5,x, algorithm="giac")

[Out]

-1/4*(4*b^3*x^3*e^3*sign(b*x + a) + 6*b^3*d*x^2*e^2*sign(b*x + a) + 4*b^3*d^2*x*
e*sign(b*x + a) + b^3*d^3*sign(b*x + a) + 6*a*b^2*x^2*e^3*sign(b*x + a) + 4*a*b^
2*d*x*e^2*sign(b*x + a) + a*b^2*d^2*e*sign(b*x + a) + 4*a^2*b*x*e^3*sign(b*x + a
) + a^2*b*d*e^2*sign(b*x + a) + a^3*e^3*sign(b*x + a))*e^(-4)/(x*e + d)^4