3.1736 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x) (d+e x)^8}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (a+b x) (d+e x)^9}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6} \]

[Out]

-((b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^5*(a + b*x)*(d
+ e*x)^10) + ((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])/(9*e^5*(a + b*x)*(d + e*x)^9) - (3*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])/(8*e^5*(a + b*x)*(d + e*x)^8) + (b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7) -
(b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^5*(a + b*x)*(d + e*x)^6)

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Rubi [A]  time = 0.617521, 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 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x) (d+e x)^8}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (a+b x) (d+e x)^9}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^11,x]

[Out]

-((b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^5*(a + b*x)*(d
+ e*x)^10) + ((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])/(9*e^5*(a + b*x)*(d + e*x)^9) - (3*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])/(8*e^5*(a + b*x)*(d + e*x)^8) + (b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7) -
(b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^5*(a + b*x)*(d + e*x)^6)

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Rubi in Sympy [A]  time = 54.2847, size = 304, normalized size = 1.02 \[ \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - 5 B a e + 2 B b d\right )}{360 e^{4} \left (d + e x\right )^{7} \left (a e - b d\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - 5 B a e + 2 B b d\right )}{2520 e^{5} \left (a + b x\right ) \left (d + e x\right )^{7}} + \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - 5 B a e + 2 B b d\right )}{360 e^{3} \left (d + e x\right )^{8} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20 e \left (d + e x\right )^{10} \left (a e - b d\right )} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (3 A b e - 5 B a e + 2 B b d\right )}{45 e^{2} \left (d + e x\right )^{9} \left (a e - b d\right )} \]

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)**11,x)

[Out]

b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(3*A*b*e - 5*B*a*e + 2*B*b*d)/(360*e**4*(d
+ e*x)**7*(a*e - b*d)) - b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(3*A*b*e - 5*B*a
*e + 2*B*b*d)/(2520*e**5*(a + b*x)*(d + e*x)**7) + b*(3*a + 3*b*x)*sqrt(a**2 + 2
*a*b*x + b**2*x**2)*(3*A*b*e - 5*B*a*e + 2*B*b*d)/(360*e**3*(d + e*x)**8*(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)/(20*e*(d +
e*x)**10*(a*e - b*d)) + (a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(3*A*b*e - 5*B*a*e
+ 2*B*b*d)/(45*e**2*(d + e*x)**9*(a*e - b*d))

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Mathematica [A]  time = 0.200496, size = 232, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (28 a^3 e^3 (9 A e+B (d+10 e x))+21 a^2 b e^2 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+3 a b^2 e \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )\right )}{2520 e^5 (a+b x) (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^11,x]

[Out]

-(Sqrt[(a + b*x)^2]*(28*a^3*e^3*(9*A*e + B*(d + 10*e*x)) + 21*a^2*b*e^2*(4*A*e*(
d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2)) + 3*a*b^2*e*(7*A*e*(d^2 + 10*d*e*
x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3)) + b^3*(3*
A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10*d^3*e*x + 45
*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4))))/(2520*e^5*(a + b*x)*(d + e*x)^10)

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Maple [A]  time = 0.017, size = 317, normalized size = 1.1 \[ -{\frac{420\,B{x}^{4}{b}^{3}{e}^{4}+360\,A{x}^{3}{b}^{3}{e}^{4}+1080\,B{x}^{3}a{b}^{2}{e}^{4}+240\,B{x}^{3}{b}^{3}d{e}^{3}+945\,A{x}^{2}a{b}^{2}{e}^{4}+135\,A{x}^{2}{b}^{3}d{e}^{3}+945\,B{x}^{2}{a}^{2}b{e}^{4}+405\,B{x}^{2}a{b}^{2}d{e}^{3}+90\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+840\,Ax{a}^{2}b{e}^{4}+210\,Axa{b}^{2}d{e}^{3}+30\,Ax{b}^{3}{d}^{2}{e}^{2}+280\,Bx{a}^{3}{e}^{4}+210\,Bx{a}^{2}bd{e}^{3}+90\,Bxa{b}^{2}{d}^{2}{e}^{2}+20\,Bx{b}^{3}{d}^{3}e+252\,A{a}^{3}{e}^{4}+84\,Ad{e}^{3}{a}^{2}b+21\,Aa{b}^{2}{d}^{2}{e}^{2}+3\,A{b}^{3}{d}^{3}e+28\,Bd{e}^{3}{a}^{3}+21\,B{a}^{2}b{d}^{2}{e}^{2}+9\,Ba{b}^{2}{d}^{3}e+2\,B{b}^{3}{d}^{4}}{2520\,{e}^{5} \left ( ex+d \right ) ^{10} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^11,x)

[Out]

-1/2520/e^5*(420*B*b^3*e^4*x^4+360*A*b^3*e^4*x^3+1080*B*a*b^2*e^4*x^3+240*B*b^3*
d*e^3*x^3+945*A*a*b^2*e^4*x^2+135*A*b^3*d*e^3*x^2+945*B*a^2*b*e^4*x^2+405*B*a*b^
2*d*e^3*x^2+90*B*b^3*d^2*e^2*x^2+840*A*a^2*b*e^4*x+210*A*a*b^2*d*e^3*x+30*A*b^3*
d^2*e^2*x+280*B*a^3*e^4*x+210*B*a^2*b*d*e^3*x+90*B*a*b^2*d^2*e^2*x+20*B*b^3*d^3*
e*x+252*A*a^3*e^4+84*A*a^2*b*d*e^3+21*A*a*b^2*d^2*e^2+3*A*b^3*d^3*e+28*B*a^3*d*e
^3+21*B*a^2*b*d^2*e^2+9*B*a*b^2*d^3*e+2*B*b^3*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^10/
(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)^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304865, size = 493, normalized size = 1.65 \[ -\frac{420 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 252 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 21 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 28 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 120 \,{\left (2 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 45 \,{\left (2 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 21 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 10 \,{\left (2 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 21 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 28 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \,{\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \]

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)^11,x, algorithm="fricas")

[Out]

-1/2520*(420*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 252*A*a^3*e^4 + 3*(3*B*a*b^2 + A*b^3)
*d^3*e + 21*(B*a^2*b + A*a*b^2)*d^2*e^2 + 28*(B*a^3 + 3*A*a^2*b)*d*e^3 + 120*(2*
B*b^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 45*(2*B*b^3*d^2*e^2 + 3*(3*B*a*b^
2 + A*b^3)*d*e^3 + 21*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 10*(2*B*b^3*d^3*e + 3*(3*B*
a*b^2 + A*b^3)*d^2*e^2 + 21*(B*a^2*b + A*a*b^2)*d*e^3 + 28*(B*a^3 + 3*A*a^2*b)*e
^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e^13*x^8 + 120*d^3*e^12*x^7 + 210*d^4
*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^
2 + 10*d^9*e^6*x + d^10*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)**11,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302231, size = 576, normalized size = 1.93 \[ -\frac{{\left (420 \, B b^{3} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 240 \, B b^{3} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 90 \, B b^{3} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 20 \, B b^{3} d^{3} x e{\rm sign}\left (b x + a\right ) + 2 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) + 1080 \, B a b^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 360 \, A b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 405 \, B a b^{2} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 135 \, A b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 90 \, B a b^{2} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 30 \, A b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 9 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) + 3 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 945 \, B a^{2} b x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 945 \, A a b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 210 \, B a^{2} b d x e^{3}{\rm sign}\left (b x + a\right ) + 210 \, A a b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 21 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 21 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 280 \, B a^{3} x e^{4}{\rm sign}\left (b x + a\right ) + 840 \, A a^{2} b x e^{4}{\rm sign}\left (b x + a\right ) + 28 \, B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 84 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 252 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{2520 \,{\left (x e + d\right )}^{10}} \]

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)^11,x, algorithm="giac")

[Out]

-1/2520*(420*B*b^3*x^4*e^4*sign(b*x + a) + 240*B*b^3*d*x^3*e^3*sign(b*x + a) + 9
0*B*b^3*d^2*x^2*e^2*sign(b*x + a) + 20*B*b^3*d^3*x*e*sign(b*x + a) + 2*B*b^3*d^4
*sign(b*x + a) + 1080*B*a*b^2*x^3*e^4*sign(b*x + a) + 360*A*b^3*x^3*e^4*sign(b*x
+ a) + 405*B*a*b^2*d*x^2*e^3*sign(b*x + a) + 135*A*b^3*d*x^2*e^3*sign(b*x + a)
+ 90*B*a*b^2*d^2*x*e^2*sign(b*x + a) + 30*A*b^3*d^2*x*e^2*sign(b*x + a) + 9*B*a*
b^2*d^3*e*sign(b*x + a) + 3*A*b^3*d^3*e*sign(b*x + a) + 945*B*a^2*b*x^2*e^4*sign
(b*x + a) + 945*A*a*b^2*x^2*e^4*sign(b*x + a) + 210*B*a^2*b*d*x*e^3*sign(b*x + a
) + 210*A*a*b^2*d*x*e^3*sign(b*x + a) + 21*B*a^2*b*d^2*e^2*sign(b*x + a) + 21*A*
a*b^2*d^2*e^2*sign(b*x + a) + 280*B*a^3*x*e^4*sign(b*x + a) + 840*A*a^2*b*x*e^4*
sign(b*x + a) + 28*B*a^3*d*e^3*sign(b*x + a) + 84*A*a^2*b*d*e^3*sign(b*x + a) +
252*A*a^3*e^4*sign(b*x + a))*e^(-5)/(x*e + d)^10