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

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]

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

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Rubi [A]  time = 0.14, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \[ -\frac {c \left (a B e^2-2 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^4}+\frac {2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6 (d+e x)^5}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^7}+\frac {c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2} \]

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)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^8}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^7}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^6}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^5}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)^4}+\frac {B c^2}{e^5 (d+e x)^3}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{7 e^6 (d+e x)^7}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{6 e^6 (d+e x)^6}+\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{5 e^6 (d+e x)^5}-\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right )}{2 e^6 (d+e x)^4}+\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}\\ \end {align*}

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Mathematica [A]  time = 0.09, 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]

-1/210*(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 + 3
5*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)))/(e^6*(d + e*x)^7)

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fricas [A]  time = 0.61, size = 317, normalized size = 1.54 \[ -\frac {105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \, {\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \, {\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \, {\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(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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giac [A]  time = 0.18, size = 242, normalized size = 1.17 \[ -\frac {{\left (105 \, B c^{2} x^{5} e^{5} + 175 \, B c^{2} d x^{4} e^{4} + 175 \, B c^{2} d^{2} x^{3} e^{3} + 105 \, B c^{2} d^{3} x^{2} e^{2} + 35 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 70 \, A c^{2} x^{4} e^{5} + 70 \, A c^{2} d x^{3} e^{4} + 42 \, A c^{2} d^{2} x^{2} e^{3} + 14 \, A c^{2} d^{3} x e^{2} + 2 \, A c^{2} d^{4} e + 105 \, B a c x^{3} e^{5} + 63 \, B a c d x^{2} e^{4} + 21 \, B a c d^{2} x e^{3} + 3 \, B a c d^{3} e^{2} + 84 \, A a c x^{2} e^{5} + 28 \, A a c d x e^{4} + 4 \, A a c d^{2} e^{3} + 35 \, B a^{2} x e^{5} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{210 \, {\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(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

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maple [A]  time = 0.05, size = 249, normalized size = 1.21 \[ -\frac {B \,c^{2}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {\left (A e -5 B d \right ) c^{2}}{3 \left (e x +d \right )^{3} e^{6}}+\frac {\left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right ) c}{2 \left (e x +d \right )^{4} e^{6}}-\frac {2 \left (a A \,e^{3}+3 A c \,d^{2} e -3 a B d \,e^{2}-5 B c \,d^{3}\right ) c}{5 \left (e x +d \right )^{5} e^{6}}-\frac {-4 A d a c \,e^{3}-4 A \,c^{2} d^{3} e +B \,a^{2} e^{4}+6 B \,d^{2} a c \,e^{2}+5 B \,d^{4} c^{2}}{6 \left (e x +d \right )^{6} e^{6}}-\frac {A \,a^{2} e^{5}+2 A \,d^{2} a c \,e^{3}+A \,c^{2} d^{4} e -B d \,a^{2} e^{4}-2 B \,d^{3} a c \,e^{2}-B \,d^{5} c^{2}}{7 \left (e x +d \right )^{7} e^{6}} \]

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]

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

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maxima [A]  time = 0.58, size = 317, normalized size = 1.54 \[ -\frac {105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \, {\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \, {\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \, {\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(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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mupad [B]  time = 1.73, size = 299, normalized size = 1.45 \[ -\frac {\frac {5\,B\,a^2\,d\,e^4+30\,A\,a^2\,e^5+3\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+2\,A\,c^2\,d^4\,e}{210\,e^6}+\frac {x\,\left (5\,B\,a^2\,e^4+3\,B\,a\,c\,d^2\,e^2+4\,A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+2\,A\,c^2\,d^3\,e\right )}{30\,e^5}+\frac {c\,x^3\,\left (5\,B\,c\,d^2+2\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{6\,e^3}+\frac {c^2\,x^4\,\left (2\,A\,e+5\,B\,d\right )}{6\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+2\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+4\,A\,a\,e^3\right )}{10\,e^4}+\frac {B\,c^2\,x^5}{2\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((30*A*a^2*e^5 + 5*B*c^2*d^5 + 5*B*a^2*d*e^4 + 2*A*c^2*d^4*e + 4*A*a*c*d^2*e^3 + 3*B*a*c*d^3*e^2)/(210*e^6) +
(x*(5*B*a^2*e^4 + 5*B*c^2*d^4 + 2*A*c^2*d^3*e + 4*A*a*c*d*e^3 + 3*B*a*c*d^2*e^2))/(30*e^5) + (c*x^3*(3*B*a*e^
2 + 5*B*c*d^2 + 2*A*c*d*e))/(6*e^3) + (c^2*x^4*(2*A*e + 5*B*d))/(6*e^2) + (c*x^2*(4*A*a*e^3 + 5*B*c*d^3 + 3*B*
a*d*e^2 + 2*A*c*d^2*e))/(10*e^4) + (B*c^2*x^5)/(2*e))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e
^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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