3.1677 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=204 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^4 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.210923, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^4 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.6246, size = 168, normalized size = 0.82 \[ \frac{16 b^{2} \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{3}} + \frac{32 b^{2} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{4} \left (a + b x\right )} - \frac{4 b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{2} \sqrt{d + e x}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

16*b**2*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**3) + 32*b**2*sqrt(d
+ e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**4*(a + b*x)) - 4*b*(3
*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**2*sqrt(d + e*x)) - 2*(a**2 +
2*a*b*x + b**2*x**2)**(3/2)/(3*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.203196, size = 92, normalized size = 0.45 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} \sqrt{d+e x} \left (b^2 (9 a e-8 b d)-\frac{9 b (b d-a e)^2}{d+e x}+\frac{(b d-a e)^3}{(d+e x)^2}+b^3 e x\right )}{3 e^4 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a + b*x)^2)^(3/2)*Sqrt[d + e*x]*(b^2*(-8*b*d + 9*a*e) + b^3*e*x + (b*d - a*
e)^3/(d + e*x)^2 - (9*b*(b*d - a*e)^2)/(d + e*x)))/(3*e^4*(a + b*x)^3)

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Maple [A]  time = 0.01, size = 131, normalized size = 0.6 \[ -{\frac{-2\,{x}^{3}{b}^{3}{e}^{3}-18\,{x}^{2}a{b}^{2}{e}^{3}+12\,{x}^{2}{b}^{3}d{e}^{2}+18\,x{a}^{2}b{e}^{3}-72\,xa{b}^{2}d{e}^{2}+48\,x{b}^{3}{d}^{2}e+2\,{a}^{3}{e}^{3}+12\,{a}^{2}bd{e}^{2}-48\,a{b}^{2}{d}^{2}e+32\,{b}^{3}{d}^{3}}{3\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

-2/3/(e*x+d)^(3/2)*(-b^3*e^3*x^3-9*a*b^2*e^3*x^2+6*b^3*d*e^2*x^2+9*a^2*b*e^3*x-3
6*a*b^2*d*e^2*x+24*b^3*d^2*e*x+a^3*e^3+6*a^2*b*d*e^2-24*a*b^2*d^2*e+16*b^3*d^3)*
((b*x+a)^2)^(3/2)/e^4/(b*x+a)^3

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Maxima [A]  time = 0.724914, size = 169, normalized size = 0.83 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )}}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 6*a^2*b*d*e^2 - a^3*e^3 - 3*(2*
b^3*d*e^2 - 3*a*b^2*e^3)*x^2 - 3*(8*b^3*d^2*e - 12*a*b^2*d*e^2 + 3*a^2*b*e^3)*x)
/((e^5*x + d*e^4)*sqrt(e*x + d))

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Fricas [A]  time = 0.206121, size = 169, normalized size = 0.83 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )}}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 6*a^2*b*d*e^2 - a^3*e^3 - 3*(2*
b^3*d*e^2 - 3*a*b^2*e^3)*x^2 - 3*(8*b^3*d^2*e - 12*a*b^2*d*e^2 + 3*a^2*b*e^3)*x)
/((e^5*x + d*e^4)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220885, size = 273, normalized size = 1.34 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{8}{\rm sign}\left (b x + a\right ) - 9 \, \sqrt{x e + d} b^{3} d e^{8}{\rm sign}\left (b x + a\right ) + 9 \, \sqrt{x e + d} a b^{2} e^{9}{\rm sign}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} b^{3} d^{2}{\rm sign}\left (b x + a\right ) - b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 18 \,{\left (x e + d\right )} a b^{2} d e{\rm sign}\left (b x + a\right ) + 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 9 \,{\left (x e + d\right )} a^{2} b e^{2}{\rm sign}\left (b x + a\right ) - 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((x*e + d)^(3/2)*b^3*e^8*sign(b*x + a) - 9*sqrt(x*e + d)*b^3*d*e^8*sign(b*x
+ a) + 9*sqrt(x*e + d)*a*b^2*e^9*sign(b*x + a))*e^(-12) - 2/3*(9*(x*e + d)*b^3*d
^2*sign(b*x + a) - b^3*d^3*sign(b*x + a) - 18*(x*e + d)*a*b^2*d*e*sign(b*x + a)
+ 3*a*b^2*d^2*e*sign(b*x + a) + 9*(x*e + d)*a^2*b*e^2*sign(b*x + a) - 3*a^2*b*d*
e^2*sign(b*x + a) + a^3*e^3*sign(b*x + a))*e^(-4)/(x*e + d)^(3/2)