3.1678 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) \sqrt{d+e x}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^4 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.202229, antiderivative size = 202, 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 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) \sqrt{d+e x}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^4 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.5415, size = 168, normalized size = 0.83 \[ \frac{16 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e^{3} \sqrt{d + e x}} - \frac{32 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e^{4} \left (a + b x\right ) \sqrt{d + e x}} - \frac{4 b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

16*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e**3*sqrt(d + e*x)) - 32*b**2*(a*e -
b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e**4*(a + b*x)*sqrt(d + e*x)) - 4*b*(3
*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**2*(d + e*x)**(3/2)) - 2*(a**
2 + 2*a*b*x + b**2*x**2)**(3/2)/(5*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.186338, size = 98, normalized size = 0.49 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} \sqrt{d+e x} \left (\frac{15 b^2 (b d-a e)}{d+e x}-\frac{5 b (b d-a e)^2}{(d+e x)^2}+\frac{(b d-a e)^3}{(d+e x)^3}+5 b^3\right )}{5 e^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

(2*((a + b*x)^2)^(3/2)*Sqrt[d + e*x]*(5*b^3 + (b*d - a*e)^3/(d + e*x)^3 - (5*b*(
b*d - a*e)^2)/(d + e*x)^2 + (15*b^2*(b*d - a*e))/(d + e*x)))/(5*e^4*(a + b*x)^3)

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Maple [A]  time = 0.01, size = 131, normalized size = 0.7 \[ -{\frac{-10\,{x}^{3}{b}^{3}{e}^{3}+30\,{x}^{2}a{b}^{2}{e}^{3}-60\,{x}^{2}{b}^{3}d{e}^{2}+10\,x{a}^{2}b{e}^{3}+40\,xa{b}^{2}d{e}^{2}-80\,x{b}^{3}{d}^{2}e+2\,{a}^{3}{e}^{3}+4\,{a}^{2}bd{e}^{2}+16\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{5\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(7/2),x)

[Out]

-2/5/(e*x+d)^(5/2)*(-5*b^3*e^3*x^3+15*a*b^2*e^3*x^2-30*b^3*d*e^2*x^2+5*a^2*b*e^3
*x+20*a*b^2*d*e^2*x-40*b^3*d^2*e*x+a^3*e^3+2*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 = 0.740461, size = 185, normalized size = 0.92 \[ \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )}}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

2/5*(5*b^3*e^3*x^3 + 16*b^3*d^3 - 8*a*b^2*d^2*e - 2*a^2*b*d*e^2 - a^3*e^3 + 15*(
2*b^3*d*e^2 - a*b^2*e^3)*x^2 + 5*(8*b^3*d^2*e - 4*a*b^2*d*e^2 - a^2*b*e^3)*x)/((
e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(e*x + d))

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Fricas [A]  time = 0.207344, size = 185, normalized size = 0.92 \[ \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )}}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/5*(5*b^3*e^3*x^3 + 16*b^3*d^3 - 8*a*b^2*d^2*e - 2*a^2*b*d*e^2 - a^3*e^3 + 15*(
2*b^3*d*e^2 - a*b^2*e^3)*x^2 + 5*(8*b^3*d^2*e - 4*a*b^2*d*e^2 - a^2*b*e^3)*x)/((
e^6*x^2 + 2*d*e^5*x + d^2*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21894, size = 265, normalized size = 1.31 \[ 2 \, \sqrt{x e + d} b^{3} e^{\left (-4\right )}{\rm sign}\left (b x + a\right ) + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} b^{3} d{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )} b^{3} d^{2}{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 15 \,{\left (x e + d\right )}^{2} a b^{2} e{\rm sign}\left (b x + a\right ) + 10 \,{\left (x e + d\right )} a b^{2} d e{\rm sign}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )} a^{2} b e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

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)^(7/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^3*e^(-4)*sign(b*x + a) + 2/5*(15*(x*e + d)^2*b^3*d*sign(b*x +
a) - 5*(x*e + d)*b^3*d^2*sign(b*x + a) + b^3*d^3*sign(b*x + a) - 15*(x*e + d)^2*
a*b^2*e*sign(b*x + a) + 10*(x*e + d)*a*b^2*d*e*sign(b*x + a) - 3*a*b^2*d^2*e*sig
n(b*x + a) - 5*(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)^(5/2)