3.1679 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=204 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) (d+e x)^{3/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^4 (a+b x) (d+e x)^{5/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^{7/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) \sqrt{d+e x}} \]

[Out]

(2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)^(7/2)
) - (6*b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)*(d + e*x)
^(5/2)) + (2*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)*(d +
e*x)^(3/2)) - (2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)*Sqrt[d + e*x]
)

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Rubi [A]  time = 0.202482, 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 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) (d+e x)^{3/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^4 (a+b x) (d+e x)^{5/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^{7/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)^(7/2)
) - (6*b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)*(d + e*x)
^(5/2)) + (2*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)*(d +
e*x)^(3/2)) - (2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)*Sqrt[d + e*x]
)

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Rubi in Sympy [A]  time = 22.8865, size = 168, normalized size = 0.82 \[ - \frac{48 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{3} \left (d + e x\right )^{\frac{3}{2}}} + \frac{32 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{4} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(9/2),x)

[Out]

-48*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**3*(d + e*x)**(3/2)) + 32*b**2*(
a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**4*(a + b*x)*(d + e*x)**(3/2))
- 4*b*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**2*(d + e*x)**(5/2))
- 2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7*e*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.126, size = 116, normalized size = 0.57 \[ \frac{\left ((a+b x)^2\right )^{3/2} \sqrt{d+e x} \left (\frac{2 b^2 (b d-a e)}{e^4 (d+e x)^2}-\frac{6 b (b d-a e)^2}{5 e^4 (d+e x)^3}-\frac{2 (a e-b d)^3}{7 e^4 (d+e x)^4}-\frac{2 b^3}{e^4 (d+e x)}\right )}{(a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(((a + b*x)^2)^(3/2)*Sqrt[d + e*x]*((-2*(-(b*d) + a*e)^3)/(7*e^4*(d + e*x)^4) -
(6*b*(b*d - a*e)^2)/(5*e^4*(d + e*x)^3) + (2*b^2*(b*d - a*e))/(e^4*(d + e*x)^2)
- (2*b^3)/(e^4*(d + e*x))))/(a + b*x)^3

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Maple [A]  time = 0.01, size = 132, normalized size = 0.7 \[ -{\frac{70\,{x}^{3}{b}^{3}{e}^{3}+70\,{x}^{2}a{b}^{2}{e}^{3}+140\,{x}^{2}{b}^{3}d{e}^{2}+42\,x{a}^{2}b{e}^{3}+56\,xa{b}^{2}d{e}^{2}+112\,x{b}^{3}{d}^{2}e+10\,{a}^{3}{e}^{3}+12\,{a}^{2}bd{e}^{2}+16\,a{b}^{2}{d}^{2}e+32\,{b}^{3}{d}^{3}}{35\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

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)^(9/2),x)

[Out]

-2/35/(e*x+d)^(7/2)*(35*b^3*e^3*x^3+35*a*b^2*e^3*x^2+70*b^3*d*e^2*x^2+21*a^2*b*e
^3*x+28*a*b^2*d*e^2*x+56*b^3*d^2*e*x+5*a^3*e^3+6*a^2*b*d*e^2+8*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 = 0.738559, size = 198, normalized size = 0.97 \[ -\frac{2 \,{\left (35 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 8 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 35 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 7 \,{\left (8 \, b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )}}{35 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )} \sqrt{e x + d}} \]

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)^(9/2),x, algorithm="maxima")

[Out]

-2/35*(35*b^3*e^3*x^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2 + 5*a^3*e^3 +
35*(2*b^3*d*e^2 + a*b^2*e^3)*x^2 + 7*(8*b^3*d^2*e + 4*a*b^2*d*e^2 + 3*a^2*b*e^3
)*x)/((e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)*sqrt(e*x + d))

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Fricas [A]  time = 0.208131, size = 198, normalized size = 0.97 \[ -\frac{2 \,{\left (35 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 8 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 35 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 7 \,{\left (8 \, b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )}}{35 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )} \sqrt{e x + d}} \]

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)^(9/2),x, algorithm="fricas")

[Out]

-2/35*(35*b^3*e^3*x^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2 + 5*a^3*e^3 +
35*(2*b^3*d*e^2 + a*b^2*e^3)*x^2 + 7*(8*b^3*d^2*e + 4*a*b^2*d*e^2 + 3*a^2*b*e^3
)*x)/((e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)*sqrt(e*x + d))

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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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220384, size = 262, normalized size = 1.28 \[ -\frac{2 \,{\left (35 \,{\left (x e + d\right )}^{3} b^{3}{\rm sign}\left (b x + a\right ) - 35 \,{\left (x e + d\right )}^{2} b^{3} d{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )} b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 5 \, b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{2} a b^{2} e{\rm sign}\left (b x + a\right ) - 42 \,{\left (x e + d\right )} a b^{2} d e{\rm sign}\left (b x + a\right ) + 15 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )} a^{2} b e^{2}{\rm sign}\left (b x + a\right ) - 15 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{35 \,{\left (x e + d\right )}^{\frac{7}{2}}} \]

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)^(9/2),x, algorithm="giac")

[Out]

-2/35*(35*(x*e + d)^3*b^3*sign(b*x + a) - 35*(x*e + d)^2*b^3*d*sign(b*x + a) + 2
1*(x*e + d)*b^3*d^2*sign(b*x + a) - 5*b^3*d^3*sign(b*x + a) + 35*(x*e + d)^2*a*b
^2*e*sign(b*x + a) - 42*(x*e + d)*a*b^2*d*e*sign(b*x + a) + 15*a*b^2*d^2*e*sign(
b*x + a) + 21*(x*e + d)*a^2*b*e^2*sign(b*x + a) - 15*a^2*b*d*e^2*sign(b*x + a) +
5*a^3*e^3*sign(b*x + a))*e^(-4)/(x*e + d)^(7/2)