3.1737 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{11 e^5 (a+b x) (d+e x)^{11}}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]

[Out]

-((b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)*(d
+ e*x)^11) + ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(10*e^5*(a + b*x)*(d + e*x)^10) - (b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^9) + (b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)*(d + e*x)^8) -
(b^3*B*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 = 0.619438, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{11 e^5 (a+b x) (d+e x)^{11}}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]

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)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)*(d
+ e*x)^11) + ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(10*e^5*(a + b*x)*(d + e*x)^10) - (b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^9) + (b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)*(d + e*x)^8) -
(b^3*B*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 = 54.2526, size = 304, normalized size = 1.02 \[ \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e - 11 B a e + 4 B b d\right )}{1155 e^{4} \left (d + e x\right )^{8} \left (a e - b d\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e - 11 B a e + 4 B b d\right )}{9240 e^{5} \left (a + b x\right ) \left (d + e x\right )^{8}} + \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e - 11 B a e + 4 B b d\right )}{990 e^{3} \left (d + e x\right )^{9} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{22 e \left (d + e x\right )^{11} \left (a e - b d\right )} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (7 A b e - 11 B a e + 4 B b d\right )}{110 e^{2} \left (d + e x\right )^{10} \left (a e - b d\right )} \]

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]

b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(7*A*b*e - 11*B*a*e + 4*B*b*d)/(1155*e**4*
(d + e*x)**8*(a*e - b*d)) - b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(7*A*b*e - 11*
B*a*e + 4*B*b*d)/(9240*e**5*(a + b*x)*(d + e*x)**8) + b*(3*a + 3*b*x)*sqrt(a**2
+ 2*a*b*x + b**2*x**2)*(7*A*b*e - 11*B*a*e + 4*B*b*d)/(990*e**3*(d + e*x)**9*(a*
e - b*d)) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(22*e*
(d + e*x)**11*(a*e - b*d)) + (a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(7*A*b*e - 11*B
*a*e + 4*B*b*d)/(110*e**2*(d + e*x)**10*(a*e - b*d))

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Mathematica [A]  time = 0.19821, size = 233, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (84 a^3 e^3 (10 A e+B (d+11 e x))+28 a^2 b e^2 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+7 a b^2 e \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+b^3 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )\right )}{9240 e^5 (a+b x) (d+e x)^{11}} \]

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]*(84*a^3*e^3*(10*A*e + B*(d + 11*e*x)) + 28*a^2*b*e^2*(9*A*e*
(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)) + 7*a*b^2*e*(8*A*e*(d^2 + 11*d
*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3)) + b^3*
(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(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))))/(9240*e^5*(a + b*x)*(d + e*x)^
11)

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Maple [A]  time = 0.016, size = 317, normalized size = 1.1 \[ -{\frac{1320\,B{x}^{4}{b}^{3}{e}^{4}+1155\,A{x}^{3}{b}^{3}{e}^{4}+3465\,B{x}^{3}a{b}^{2}{e}^{4}+660\,B{x}^{3}{b}^{3}d{e}^{3}+3080\,A{x}^{2}a{b}^{2}{e}^{4}+385\,A{x}^{2}{b}^{3}d{e}^{3}+3080\,B{x}^{2}{a}^{2}b{e}^{4}+1155\,B{x}^{2}a{b}^{2}d{e}^{3}+220\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+2772\,Ax{a}^{2}b{e}^{4}+616\,Axa{b}^{2}d{e}^{3}+77\,Ax{b}^{3}{d}^{2}{e}^{2}+924\,Bx{a}^{3}{e}^{4}+616\,Bx{a}^{2}bd{e}^{3}+231\,Bxa{b}^{2}{d}^{2}{e}^{2}+44\,Bx{b}^{3}{d}^{3}e+840\,A{a}^{3}{e}^{4}+252\,Ad{e}^{3}{a}^{2}b+56\,Aa{b}^{2}{d}^{2}{e}^{2}+7\,A{b}^{3}{d}^{3}e+84\,Bd{e}^{3}{a}^{3}+56\,B{a}^{2}b{d}^{2}{e}^{2}+21\,Ba{b}^{2}{d}^{3}e+4\,B{b}^{3}{d}^{4}}{9240\,{e}^{5} \left ( ex+d \right ) ^{11} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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]

-1/9240/e^5*(1320*B*b^3*e^4*x^4+1155*A*b^3*e^4*x^3+3465*B*a*b^2*e^4*x^3+660*B*b^
3*d*e^3*x^3+3080*A*a*b^2*e^4*x^2+385*A*b^3*d*e^3*x^2+3080*B*a^2*b*e^4*x^2+1155*B
*a*b^2*d*e^3*x^2+220*B*b^3*d^2*e^2*x^2+2772*A*a^2*b*e^4*x+616*A*a*b^2*d*e^3*x+77
*A*b^3*d^2*e^2*x+924*B*a^3*e^4*x+616*B*a^2*b*d*e^3*x+231*B*a*b^2*d^2*e^2*x+44*B*
b^3*d^3*e*x+840*A*a^3*e^4+252*A*a^2*b*d*e^3+56*A*a*b^2*d^2*e^2+7*A*b^3*d^3*e+84*
B*a^3*d*e^3+56*B*a^2*b*d^2*e^2+21*B*a*b^2*d^3*e+4*B*b^3*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 = 0.276048, size = 508, normalized size = 1.7 \[ -\frac{1320 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 840 \, A a^{3} e^{4} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 56 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 84 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 165 \,{\left (4 \, B b^{3} d e^{3} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 55 \,{\left (4 \, B b^{3} d^{2} e^{2} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 56 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 11 \,{\left (4 \, B b^{3} d^{3} e + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 56 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 84 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{9240 \,{\left (e^{16} x^{11} + 11 \, d e^{15} x^{10} + 55 \, d^{2} e^{14} x^{9} + 165 \, d^{3} e^{13} x^{8} + 330 \, d^{4} e^{12} x^{7} + 462 \, d^{5} e^{11} x^{6} + 462 \, d^{6} e^{10} x^{5} + 330 \, d^{7} e^{9} x^{4} + 165 \, d^{8} e^{8} x^{3} + 55 \, d^{9} e^{7} x^{2} + 11 \, d^{10} e^{6} x + d^{11} e^{5}\right )}} \]

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]

-1/9240*(1320*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 840*A*a^3*e^4 + 7*(3*B*a*b^2 + A*b^3
)*d^3*e + 56*(B*a^2*b + A*a*b^2)*d^2*e^2 + 84*(B*a^3 + 3*A*a^2*b)*d*e^3 + 165*(4
*B*b^3*d*e^3 + 7*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 55*(4*B*b^3*d^2*e^2 + 7*(3*B*a*b
^2 + A*b^3)*d*e^3 + 56*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 11*(4*B*b^3*d^3*e + 7*(3*B
*a*b^2 + A*b^3)*d^2*e^2 + 56*(B*a^2*b + A*a*b^2)*d*e^3 + 84*(B*a^3 + 3*A*a^2*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 = 0.3016, size = 576, normalized size = 1.93 \[ -\frac{{\left (1320 \, B b^{3} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 660 \, B b^{3} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 220 \, B b^{3} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 44 \, B b^{3} d^{3} x e{\rm sign}\left (b x + a\right ) + 4 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) + 3465 \, B a b^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 1155 \, A b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 1155 \, B a b^{2} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 385 \, A b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 231 \, B a b^{2} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 77 \, A b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 21 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) + 7 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 3080 \, B a^{2} b x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 3080 \, A a b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 616 \, B a^{2} b d x e^{3}{\rm sign}\left (b x + a\right ) + 616 \, A a b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 56 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 56 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 924 \, B a^{3} x e^{4}{\rm sign}\left (b x + a\right ) + 2772 \, A a^{2} b x e^{4}{\rm sign}\left (b x + a\right ) + 84 \, B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 252 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 840 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{9240 \,{\left (x e + d\right )}^{11}} \]

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]

-1/9240*(1320*B*b^3*x^4*e^4*sign(b*x + a) + 660*B*b^3*d*x^3*e^3*sign(b*x + a) +
220*B*b^3*d^2*x^2*e^2*sign(b*x + a) + 44*B*b^3*d^3*x*e*sign(b*x + a) + 4*B*b^3*d
^4*sign(b*x + a) + 3465*B*a*b^2*x^3*e^4*sign(b*x + a) + 1155*A*b^3*x^3*e^4*sign(
b*x + a) + 1155*B*a*b^2*d*x^2*e^3*sign(b*x + a) + 385*A*b^3*d*x^2*e^3*sign(b*x +
a) + 231*B*a*b^2*d^2*x*e^2*sign(b*x + a) + 77*A*b^3*d^2*x*e^2*sign(b*x + a) + 2
1*B*a*b^2*d^3*e*sign(b*x + a) + 7*A*b^3*d^3*e*sign(b*x + a) + 3080*B*a^2*b*x^2*e
^4*sign(b*x + a) + 3080*A*a*b^2*x^2*e^4*sign(b*x + a) + 616*B*a^2*b*d*x*e^3*sign
(b*x + a) + 616*A*a*b^2*d*x*e^3*sign(b*x + a) + 56*B*a^2*b*d^2*e^2*sign(b*x + a)
+ 56*A*a*b^2*d^2*e^2*sign(b*x + a) + 924*B*a^3*x*e^4*sign(b*x + a) + 2772*A*a^2
*b*x*e^4*sign(b*x + a) + 84*B*a^3*d*e^3*sign(b*x + a) + 252*A*a^2*b*d*e^3*sign(b
*x + a) + 840*A*a^3*e^4*sign(b*x + a))*e^(-5)/(x*e + d)^11