3.1987 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^5 (a+b x) (d+e x)^9}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^5 (a+b x) (d+e x)^{11}}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^5 (a+b x) (d+e x)^8} \]

[Out]

-((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)*(d + e*x)^11) +
(2*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^10
) - (2*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*
x)^9) + (b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)*(d + e*
x)^8) - (b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7)

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Rubi [A]  time = 0.376957, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^5 (a+b x) (d+e x)^9}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^5 (a+b x) (d+e x)^{11}}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^5 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^12,x]

[Out]

-((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)*(d + e*x)^11) +
(2*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^10
) - (2*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*
x)^9) + (b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)*(d + e*
x)^8) - (b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7)

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Rubi in Sympy [A]  time = 37.6349, size = 201, normalized size = 0.79 \[ - \frac{4 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{1155 e^{4} \left (d + e x\right )^{8}} + \frac{b^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2310 e^{5} \left (a + b x\right ) \left (d + e x\right )^{8}} - \frac{2 b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{495 e^{3} \left (d + e x\right )^{9}} - \frac{2 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{55 e^{2} \left (d + e x\right )^{10}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{11 e \left (d + e x\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**12,x)

[Out]

-4*b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(1155*e**4*(d + e*x)**8) + b**3*(a*e -
b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2310*e**5*(a + b*x)*(d + e*x)**8) - 2*b**
2*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(495*e**3*(d + e*x)**9) - 2*b*(
a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(55*e**2*(d + e*x)**10) - (a + b*x)*(a**2 + 2
*a*b*x + b**2*x**2)**(3/2)/(11*e*(d + e*x)**11)

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Mathematica [A]  time = 0.12855, size = 162, normalized size = 0.64 \[ -\frac{\sqrt{(a+b x)^2} \left (210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+7 a b^3 e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )}{2310 e^5 (a+b x) (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(210*a^4*e^4 + 84*a^3*b*e^3*(d + 11*e*x) + 28*a^2*b^2*e^2*(d
^2 + 11*d*e*x + 55*e^2*x^2) + 7*a*b^3*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e
^3*x^3) + b^4*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4))
)/(2310*e^5*(a + b*x)*(d + e*x)^11)

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Maple [A]  time = 0.016, size = 201, normalized size = 0.8 \[ -{\frac{330\,{x}^{4}{b}^{4}{e}^{4}+1155\,{x}^{3}a{b}^{3}{e}^{4}+165\,{x}^{3}{b}^{4}d{e}^{3}+1540\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+385\,{x}^{2}a{b}^{3}d{e}^{3}+55\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+924\,x{a}^{3}b{e}^{4}+308\,x{a}^{2}{b}^{2}d{e}^{3}+77\,xa{b}^{3}{d}^{2}{e}^{2}+11\,x{b}^{4}{d}^{3}e+210\,{a}^{4}{e}^{4}+84\,{a}^{3}bd{e}^{3}+28\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+7\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{2310\,{e}^{5} \left ( ex+d \right ) ^{11} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^12,x)

[Out]

-1/2310/e^5*(330*b^4*e^4*x^4+1155*a*b^3*e^4*x^3+165*b^4*d*e^3*x^3+1540*a^2*b^2*e
^4*x^2+385*a*b^3*d*e^3*x^2+55*b^4*d^2*e^2*x^2+924*a^3*b*e^4*x+308*a^2*b^2*d*e^3*
x+77*a*b^3*d^2*e^2*x+11*b^4*d^3*e*x+210*a^4*e^4+84*a^3*b*d*e^3+28*a^2*b^2*d^2*e^
2+7*a*b^3*d^3*e+b^4*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^11/(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)*(b*x + a)/(e*x + d)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290529, size = 393, normalized size = 1.55 \[ -\frac{330 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 7 \, a b^{3} d^{3} e + 28 \, a^{2} b^{2} d^{2} e^{2} + 84 \, a^{3} b d e^{3} + 210 \, a^{4} e^{4} + 165 \,{\left (b^{4} d e^{3} + 7 \, a b^{3} e^{4}\right )} x^{3} + 55 \,{\left (b^{4} d^{2} e^{2} + 7 \, a b^{3} d e^{3} + 28 \, a^{2} b^{2} e^{4}\right )} x^{2} + 11 \,{\left (b^{4} d^{3} e + 7 \, a b^{3} d^{2} e^{2} + 28 \, a^{2} b^{2} d e^{3} + 84 \, a^{3} b e^{4}\right )} x}{2310 \,{\left (e^{16} x^{11} + 11 \, d e^{15} x^{10} + 55 \, d^{2} e^{14} x^{9} + 165 \, d^{3} e^{13} x^{8} + 330 \, d^{4} e^{12} x^{7} + 462 \, d^{5} e^{11} x^{6} + 462 \, d^{6} e^{10} x^{5} + 330 \, d^{7} e^{9} x^{4} + 165 \, d^{8} e^{8} x^{3} + 55 \, d^{9} e^{7} x^{2} + 11 \, d^{10} e^{6} x + d^{11} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^12,x, algorithm="fricas")

[Out]

-1/2310*(330*b^4*e^4*x^4 + b^4*d^4 + 7*a*b^3*d^3*e + 28*a^2*b^2*d^2*e^2 + 84*a^3
*b*d*e^3 + 210*a^4*e^4 + 165*(b^4*d*e^3 + 7*a*b^3*e^4)*x^3 + 55*(b^4*d^2*e^2 + 7
*a*b^3*d*e^3 + 28*a^2*b^2*e^4)*x^2 + 11*(b^4*d^3*e + 7*a*b^3*d^2*e^2 + 28*a^2*b^
2*d*e^3 + 84*a^3*b*e^4)*x)/(e^16*x^11 + 11*d*e^15*x^10 + 55*d^2*e^14*x^9 + 165*d
^3*e^13*x^8 + 330*d^4*e^12*x^7 + 462*d^5*e^11*x^6 + 462*d^6*e^10*x^5 + 330*d^7*e
^9*x^4 + 165*d^8*e^8*x^3 + 55*d^9*e^7*x^2 + 11*d^10*e^6*x + d^11*e^5)

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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*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279574, size = 356, normalized size = 1.4 \[ -\frac{{\left (330 \, b^{4} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 165 \, b^{4} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 55 \, b^{4} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 11 \, b^{4} d^{3} x e{\rm sign}\left (b x + a\right ) + b^{4} d^{4}{\rm sign}\left (b x + a\right ) + 1155 \, a b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 385 \, a b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 77 \, a b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 7 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 1540 \, a^{2} b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 308 \, a^{2} b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 28 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 924 \, a^{3} b x e^{4}{\rm sign}\left (b x + a\right ) + 84 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + 210 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{2310 \,{\left (x e + d\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^12,x, algorithm="giac")

[Out]

-1/2310*(330*b^4*x^4*e^4*sign(b*x + a) + 165*b^4*d*x^3*e^3*sign(b*x + a) + 55*b^
4*d^2*x^2*e^2*sign(b*x + a) + 11*b^4*d^3*x*e*sign(b*x + a) + b^4*d^4*sign(b*x +
a) + 1155*a*b^3*x^3*e^4*sign(b*x + a) + 385*a*b^3*d*x^2*e^3*sign(b*x + a) + 77*a
*b^3*d^2*x*e^2*sign(b*x + a) + 7*a*b^3*d^3*e*sign(b*x + a) + 1540*a^2*b^2*x^2*e^
4*sign(b*x + a) + 308*a^2*b^2*d*x*e^3*sign(b*x + a) + 28*a^2*b^2*d^2*e^2*sign(b*
x + a) + 924*a^3*b*x*e^4*sign(b*x + a) + 84*a^3*b*d*e^3*sign(b*x + a) + 210*a^4*
e^4*sign(b*x + a))*e^(-5)/(x*e + d)^11