3.1310 \(\int \frac{(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=206 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^4}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^7}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6 (d+e x)^5}+\frac{c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2} \]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(7*e^6*(d + e*x)^7) - ((c*d^2 + a*e^2)*(5*B*c*d^
2 - 4*A*c*d*e + a*B*e^2))/(6*e^6*(d + e*x)^6) + (2*c*(5*B*c*d^3 - 3*A*c*d^2*e +
3*a*B*d*e^2 - a*A*e^3))/(5*e^6*(d + e*x)^5) - (c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^
2))/(2*e^6*(d + e*x)^4) + (c^2*(5*B*d - A*e))/(3*e^6*(d + e*x)^3) - (B*c^2)/(2*e
^6*(d + e*x)^2)

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Rubi [A]  time = 0.412251, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^4}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^7}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6 (d+e x)^5}+\frac{c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^8,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(7*e^6*(d + e*x)^7) - ((c*d^2 + a*e^2)*(5*B*c*d^
2 - 4*A*c*d*e + a*B*e^2))/(6*e^6*(d + e*x)^6) + (2*c*(5*B*c*d^3 - 3*A*c*d^2*e +
3*a*B*d*e^2 - a*A*e^3))/(5*e^6*(d + e*x)^5) - (c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^
2))/(2*e^6*(d + e*x)^4) + (c^2*(5*B*d - A*e))/(3*e^6*(d + e*x)^3) - (B*c^2)/(2*e
^6*(d + e*x)^2)

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Rubi in Sympy [A]  time = 69.426, size = 206, normalized size = 1. \[ - \frac{B c^{2}}{2 e^{6} \left (d + e x\right )^{2}} - \frac{c^{2} \left (A e - 5 B d\right )}{3 e^{6} \left (d + e x\right )^{3}} - \frac{c \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{2 e^{6} \left (d + e x\right )^{4}} - \frac{2 c \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{5 e^{6} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{6 e^{6} \left (d + e x\right )^{6}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{7 e^{6} \left (d + e x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**8,x)

[Out]

-B*c**2/(2*e**6*(d + e*x)**2) - c**2*(A*e - 5*B*d)/(3*e**6*(d + e*x)**3) - c*(-2
*A*c*d*e + B*a*e**2 + 5*B*c*d**2)/(2*e**6*(d + e*x)**4) - 2*c*(A*a*e**3 + 3*A*c*
d**2*e - 3*B*a*d*e**2 - 5*B*c*d**3)/(5*e**6*(d + e*x)**5) - (a*e**2 + c*d**2)*(-
4*A*c*d*e + B*a*e**2 + 5*B*c*d**2)/(6*e**6*(d + e*x)**6) - (A*e - B*d)*(a*e**2 +
c*d**2)**2/(7*e**6*(d + e*x)**7)

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Mathematica [A]  time = 0.218459, size = 202, normalized size = 0.98 \[ -\frac{2 A e \left (15 a^2 e^4+2 a c e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (5 a^2 e^4 (d+7 e x)+3 a c e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+5 c^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^8,x]

[Out]

-(2*A*e*(15*a^2*e^4 + 2*a*c*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^2*(d^4 + 7*d^3*
e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + B*(5*a^2*e^4*(d + 7*e*x) +
3*a*c*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 5*c^2*(d^5 + 7*d^4*e*x
+ 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(210*e^6*(d +
e*x)^7)

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Maple [A]  time = 0.01, size = 249, normalized size = 1.2 \[ -{\frac{-4\,Adac{e}^{3}-4\,A{c}^{2}{d}^{3}e+B{e}^{4}{a}^{2}+6\,aBc{d}^{2}{e}^{2}+5\,B{c}^{2}{d}^{4}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}+{\frac{c \left ( 2\,Acde-aB{e}^{2}-5\,Bc{d}^{2} \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2} \left ( Ae-5\,Bd \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{B{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{2\,c \left ( aA{e}^{3}+3\,Ac{d}^{2}e-3\,aBd{e}^{2}-5\,Bc{d}^{3} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{2}{e}^{5}+2\,A{d}^{2}ac{e}^{3}+A{d}^{4}{c}^{2}e-Bd{a}^{2}{e}^{4}-2\,aBc{d}^{3}{e}^{2}-B{c}^{2}{d}^{5}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^2/(e*x+d)^8,x)

[Out]

-1/6*(-4*A*a*c*d*e^3-4*A*c^2*d^3*e+B*a^2*e^4+6*B*a*c*d^2*e^2+5*B*c^2*d^4)/e^6/(e
*x+d)^6+1/2*c*(2*A*c*d*e-B*a*e^2-5*B*c*d^2)/e^6/(e*x+d)^4-1/3*c^2*(A*e-5*B*d)/e^
6/(e*x+d)^3-1/2*B*c^2/e^6/(e*x+d)^2-2/5*c*(A*a*e^3+3*A*c*d^2*e-3*B*a*d*e^2-5*B*c
*d^3)/e^6/(e*x+d)^5-1/7*(A*a^2*e^5+2*A*a*c*d^2*e^3+A*c^2*d^4*e-B*a^2*d*e^4-2*B*a
*c*d^3*e^2-B*c^2*d^5)/e^6/(e*x+d)^7

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Maxima [A]  time = 0.733111, size = 428, normalized size = 2.08 \[ -\frac{105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \,{\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \,{\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \,{\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^8,x, algorithm="maxima")

[Out]

-1/210*(105*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 2*A*c^2*d^4*e + 3*B*a*c*d^3*e^2 + 4*A*
a*c*d^2*e^3 + 5*B*a^2*d*e^4 + 30*A*a^2*e^5 + 35*(5*B*c^2*d*e^4 + 2*A*c^2*e^5)*x^
4 + 35*(5*B*c^2*d^2*e^3 + 2*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 21*(5*B*c^2*d^3*e^2
+ 2*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + 4*A*a*c*e^5)*x^2 + 7*(5*B*c^2*d^4*e + 2*A*c
^2*d^3*e^2 + 3*B*a*c*d^2*e^3 + 4*A*a*c*d*e^4 + 5*B*a^2*e^5)*x)/(e^13*x^7 + 7*d*e
^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 +
7*d^6*e^7*x + d^7*e^6)

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Fricas [A]  time = 0.273347, size = 428, normalized size = 2.08 \[ -\frac{105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \,{\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \,{\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \,{\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/210*(105*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 2*A*c^2*d^4*e + 3*B*a*c*d^3*e^2 + 4*A*
a*c*d^2*e^3 + 5*B*a^2*d*e^4 + 30*A*a^2*e^5 + 35*(5*B*c^2*d*e^4 + 2*A*c^2*e^5)*x^
4 + 35*(5*B*c^2*d^2*e^3 + 2*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 21*(5*B*c^2*d^3*e^2
+ 2*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + 4*A*a*c*e^5)*x^2 + 7*(5*B*c^2*d^4*e + 2*A*c
^2*d^3*e^2 + 3*B*a*c*d^2*e^3 + 4*A*a*c*d*e^4 + 5*B*a^2*e^5)*x)/(e^13*x^7 + 7*d*e
^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 +
7*d^6*e^7*x + d^7*e^6)

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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)*(c*x**2+a)**2/(e*x+d)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.292317, size = 327, normalized size = 1.59 \[ -\frac{{\left (105 \, B c^{2} x^{5} e^{5} + 175 \, B c^{2} d x^{4} e^{4} + 175 \, B c^{2} d^{2} x^{3} e^{3} + 105 \, B c^{2} d^{3} x^{2} e^{2} + 35 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 70 \, A c^{2} x^{4} e^{5} + 70 \, A c^{2} d x^{3} e^{4} + 42 \, A c^{2} d^{2} x^{2} e^{3} + 14 \, A c^{2} d^{3} x e^{2} + 2 \, A c^{2} d^{4} e + 105 \, B a c x^{3} e^{5} + 63 \, B a c d x^{2} e^{4} + 21 \, B a c d^{2} x e^{3} + 3 \, B a c d^{3} e^{2} + 84 \, A a c x^{2} e^{5} + 28 \, A a c d x e^{4} + 4 \, A a c d^{2} e^{3} + 35 \, B a^{2} x e^{5} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{210 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/210*(105*B*c^2*x^5*e^5 + 175*B*c^2*d*x^4*e^4 + 175*B*c^2*d^2*x^3*e^3 + 105*B*
c^2*d^3*x^2*e^2 + 35*B*c^2*d^4*x*e + 5*B*c^2*d^5 + 70*A*c^2*x^4*e^5 + 70*A*c^2*d
*x^3*e^4 + 42*A*c^2*d^2*x^2*e^3 + 14*A*c^2*d^3*x*e^2 + 2*A*c^2*d^4*e + 105*B*a*c
*x^3*e^5 + 63*B*a*c*d*x^2*e^4 + 21*B*a*c*d^2*x*e^3 + 3*B*a*c*d^3*e^2 + 84*A*a*c*
x^2*e^5 + 28*A*a*c*d*x*e^4 + 4*A*a*c*d^2*e^3 + 35*B*a^2*x*e^5 + 5*B*a^2*d*e^4 +
30*A*a^2*e^5)*e^(-6)/(x*e + d)^7