3.1556 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=200 \[ \frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.243657, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.7726, size = 155, normalized size = 0.78 \[ - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{28 e^{3} \left (d + e x\right )^{5}} + \frac{b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 e^{4} \left (a + b x\right ) \left (d + e x\right )^{5}} - \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{42 e^{2} \left (d + e x\right )^{6}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e \left (d + e x\right )^{7}} \]

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)**8,x)

[Out]

-b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(28*e**3*(d + e*x)**5) + b**2*(a*e - b*d)
*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(140*e**4*(a + b*x)*(d + e*x)**5) - b*(3*a + 3
*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(42*e**2*(d + e*x)**6) - (a**2 + 2*a*b*x
+ b**2*x**2)**(3/2)/(7*e*(d + e*x)**7)

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Mathematica [A]  time = 0.078816, size = 112, normalized size = 0.56 \[ -\frac{\sqrt{(a+b x)^2} \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{140 e^4 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(20*a^3*e^3 + 10*a^2*b*e^2*(d + 7*e*x) + 4*a*b^2*e*(d^2 + 7*
d*e*x + 21*e^2*x^2) + b^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)))/(140*e
^4*(a + b*x)*(d + e*x)^7)

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Maple [A]  time = 0.013, size = 131, normalized size = 0.7 \[ -{\frac{35\,{x}^{3}{b}^{3}{e}^{3}+84\,{x}^{2}a{b}^{2}{e}^{3}+21\,{x}^{2}{b}^{3}d{e}^{2}+70\,x{a}^{2}b{e}^{3}+28\,xa{b}^{2}d{e}^{2}+7\,x{b}^{3}{d}^{2}e+20\,{a}^{3}{e}^{3}+10\,{a}^{2}bd{e}^{2}+4\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{140\,{e}^{4} \left ( ex+d \right ) ^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^8,x)

[Out]

-1/140/e^4*(35*b^3*e^3*x^3+84*a*b^2*e^3*x^2+21*b^3*d*e^2*x^2+70*a^2*b*e^3*x+28*a
*b^2*d*e^2*x+7*b^3*d^2*e*x+20*a^3*e^3+10*a^2*b*d*e^2+4*a*b^2*d^2*e+b^3*d^3)*((b*
x+a)^2)^(3/2)/(e*x+d)^7/(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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208936, size = 246, normalized size = 1.23 \[ -\frac{35 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 4 \, a b^{2} d^{2} e + 10 \, a^{2} b d e^{2} + 20 \, a^{3} e^{3} + 21 \,{\left (b^{3} d e^{2} + 4 \, a b^{2} e^{3}\right )} x^{2} + 7 \,{\left (b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 10 \, a^{2} b e^{3}\right )} x}{140 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*e^3*x^3 + b^3*d^3 + 4*a*b^2*d^2*e + 10*a^2*b*d*e^2 + 20*a^3*e^3 +
21*(b^3*d*e^2 + 4*a*b^2*e^3)*x^2 + 7*(b^3*d^2*e + 4*a*b^2*d*e^2 + 10*a^2*b*e^3)
*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3
+ 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*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)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214479, size = 228, normalized size = 1.14 \[ -\frac{{\left (35 \, b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 21 \, b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 7 \, b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 84 \, a b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 28 \, a b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 70 \, a^{2} b x e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 20 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{140 \,{\left (x e + d\right )}^{7}} \]

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)^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*x^3*e^3*sign(b*x + a) + 21*b^3*d*x^2*e^2*sign(b*x + a) + 7*b^3*d^
2*x*e*sign(b*x + a) + b^3*d^3*sign(b*x + a) + 84*a*b^2*x^2*e^3*sign(b*x + a) + 2
8*a*b^2*d*x*e^2*sign(b*x + a) + 4*a*b^2*d^2*e*sign(b*x + a) + 70*a^2*b*x*e^3*sig
n(b*x + a) + 10*a^2*b*d*e^2*sign(b*x + a) + 20*a^3*e^3*sign(b*x + a))*e^(-4)/(x*
e + d)^7