3.1554 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=98 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{20 (d+e x)^4 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{5 (d+e x)^5 (b d-a e)} \]

[Out]

((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(b*d - a*e)*(d + e*x)^5) + (b*(a
+ b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(20*(b*d - a*e)^2*(d + e*x)^4)

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Rubi [A]  time = 0.119316, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{20 (d+e x)^4 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{5 (d+e x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(b*d - a*e)*(d + e*x)^5) + (b*(a
+ b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(20*(b*d - a*e)^2*(d + e*x)^4)

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Rubi in Sympy [A]  time = 10.6721, size = 82, normalized size = 0.84 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{20 \left (d + e x\right )^{5} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 \left (d + e x\right )^{5} \left (a e - b d\right )} \]

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)**6,x)

[Out]

e*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(20*(d + e*x)**5*(a*e - b*d)**2) - (2*a +
2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*(d + e*x)**5*(a*e - b*d))

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Mathematica [A]  time = 0.094747, size = 112, normalized size = 1.14 \[ -\frac{\sqrt{(a+b x)^2} \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{20 e^4 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*
e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)))/(20*e^4*
(a + b*x)*(d + e*x)^5)

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Maple [A]  time = 0.011, size = 131, normalized size = 1.3 \[ -{\frac{10\,{x}^{3}{b}^{3}{e}^{3}+20\,{x}^{2}a{b}^{2}{e}^{3}+10\,{x}^{2}{b}^{3}d{e}^{2}+15\,x{a}^{2}b{e}^{3}+10\,xa{b}^{2}d{e}^{2}+5\,x{b}^{3}{d}^{2}e+4\,{a}^{3}{e}^{3}+3\,{a}^{2}bd{e}^{2}+2\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{20\,{e}^{4} \left ( ex+d \right ) ^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^6,x)

[Out]

-1/20/e^4*(10*b^3*e^3*x^3+20*a*b^2*e^3*x^2+10*b^3*d*e^2*x^2+15*a^2*b*e^3*x+10*a*
b^2*d*e^2*x+5*b^3*d^2*e*x+4*a^3*e^3+3*a^2*b*d*e^2+2*a*b^2*d^2*e+b^3*d^3)*((b*x+a
)^2)^(3/2)/(e*x+d)^5/(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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208176, size = 216, normalized size = 2.2 \[ -\frac{10 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 2 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 4 \, a^{3} e^{3} + 10 \,{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216921, size = 228, normalized size = 2.33 \[ -\frac{{\left (10 \, b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 10 \, b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 20 \, a b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b x e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{20 \,{\left (x e + d\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

-1/20*(10*b^3*x^3*e^3*sign(b*x + a) + 10*b^3*d*x^2*e^2*sign(b*x + a) + 5*b^3*d^2
*x*e*sign(b*x + a) + b^3*d^3*sign(b*x + a) + 20*a*b^2*x^2*e^3*sign(b*x + a) + 10
*a*b^2*d*x*e^2*sign(b*x + a) + 2*a*b^2*d^2*e*sign(b*x + a) + 15*a^2*b*x*e^3*sign
(b*x + a) + 3*a^2*b*d*e^2*sign(b*x + a) + 4*a^3*e^3*sign(b*x + a))*e^(-4)/(x*e +
d)^5