3.12.54 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=334 \[ \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8 (d+e x)^6}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{8 e^8 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^9}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8 (d+e x)^7}+\frac {c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2} \]

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Rubi [A]  time = 0.26, antiderivative size = 334, 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} \begin {gather*} \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8 (d+e x)^6}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8 (d+e x)^5}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8 (d+e x)^7}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{8 e^8 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^9}+\frac {c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^10,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(9*e^8*(d + e*x)^9) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(8
*e^8*(d + e*x)^8) + (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(7*e^8*(d + e*x)^7
) + (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(6*e^8*(d + e*x)^6) + (c
^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(5*e^8*(d + e*x)^5) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e
+ a*B*e^2))/(4*e^8*(d + e*x)^4) + (c^3*(7*B*d - A*e))/(3*e^8*(d + e*x)^3) - (B*c^3)/(2*e^8*(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 )^3}{(d+e x)^{10}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{10}}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^9}+\frac {3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^8}-\frac {c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 (d+e x)^7}+\frac {c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)^6}-\frac {3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right )}{e^7 (d+e x)^5}+\frac {c^3 (-7 B d+A e)}{e^7 (d+e x)^4}+\frac {B c^3}{e^7 (d+e x)^3}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{9 e^8 (d+e x)^9}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{8 e^8 (d+e x)^8}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{7 e^8 (d+e x)^7}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{6 e^8 (d+e x)^6}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{5 e^8 (d+e x)^5}-\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{4 e^8 (d+e x)^4}+\frac {c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 359, normalized size = 1.07 \begin {gather*} -\frac {2 A e \left (140 a^3 e^6+15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+5 B \left (7 a^3 e^6 (d+9 e x)+3 a^2 c e^4 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+3 a c^2 e^2 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+7 c^3 \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^10,x]

[Out]

-1/2520*(2*A*e*(140*a^3*e^6 + 15*a^2*c*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 9*d^3*e*x + 36*d^
2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e
^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 5*B*(7*a^3*e^6*(d + 9*e*x) + 3*a^2*c*e^4*(d^3 + 9*d^2*e*x + 36*d*e^2*x
^2 + 84*e^3*x^3) + 3*a*c^2*e^2*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^
5) + 7*c^3*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x
^6 + 36*e^7*x^7)))/(e^8*(d + e*x)^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^10,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^10, x]

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fricas [A]  time = 0.39, size = 546, normalized size = 1.63 \begin {gather*} -\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, A c^{3} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 12 \, A a c^{2} e^{7}\right )} x^{4} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} + 12 \, A a c^{2} d e^{6} + 15 \, B a^{2} c e^{7}\right )} x^{3} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} + 12 \, A a c^{2} d^{2} e^{5} + 15 \, B a^{2} c d e^{6} + 30 \, A a^{2} c e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} + 12 \, A a c^{2} d^{3} e^{4} + 15 \, B a^{2} c d^{2} e^{5} + 30 \, A a^{2} c d e^{6} + 35 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 10*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 + 12*A*a*c^2*d^4*e^3 + 15*B*a
^2*c*d^3*e^4 + 30*A*a^2*c*d^2*e^5 + 35*B*a^3*d*e^6 + 280*A*a^3*e^7 + 420*(7*B*c^3*d*e^6 + 2*A*c^3*e^7)*x^6 + 6
30*(7*B*c^3*d^2*e^5 + 2*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 126*(35*B*c^3*d^3*e^4 + 10*A*c^3*d^2*e^5 + 15*B*a*c
^2*d*e^6 + 12*A*a*c^2*e^7)*x^4 + 84*(35*B*c^3*d^4*e^3 + 10*A*c^3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 + 12*A*a*c^2*d*e
^6 + 15*B*a^2*c*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*A*c^3*d^4*e^3 + 15*B*a*c^2*d^3*e^4 + 12*A*a*c^2*d^2*e^5 +
15*B*a^2*c*d*e^6 + 30*A*a^2*c*e^7)*x^2 + 9*(35*B*c^3*d^6*e + 10*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 + 12*A*a*c
^2*d^3*e^4 + 15*B*a^2*c*d^2*e^5 + 30*A*a^2*c*d*e^6 + 35*B*a^3*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x
^7 + 84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x +
d^9*e^8)

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giac [A]  time = 0.20, size = 457, normalized size = 1.37 \begin {gather*} -\frac {{\left (1260 \, B c^{3} x^{7} e^{7} + 2940 \, B c^{3} d x^{6} e^{6} + 4410 \, B c^{3} d^{2} x^{5} e^{5} + 4410 \, B c^{3} d^{3} x^{4} e^{4} + 2940 \, B c^{3} d^{4} x^{3} e^{3} + 1260 \, B c^{3} d^{5} x^{2} e^{2} + 315 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 840 \, A c^{3} x^{6} e^{7} + 1260 \, A c^{3} d x^{5} e^{6} + 1260 \, A c^{3} d^{2} x^{4} e^{5} + 840 \, A c^{3} d^{3} x^{3} e^{4} + 360 \, A c^{3} d^{4} x^{2} e^{3} + 90 \, A c^{3} d^{5} x e^{2} + 10 \, A c^{3} d^{6} e + 1890 \, B a c^{2} x^{5} e^{7} + 1890 \, B a c^{2} d x^{4} e^{6} + 1260 \, B a c^{2} d^{2} x^{3} e^{5} + 540 \, B a c^{2} d^{3} x^{2} e^{4} + 135 \, B a c^{2} d^{4} x e^{3} + 15 \, B a c^{2} d^{5} e^{2} + 1512 \, A a c^{2} x^{4} e^{7} + 1008 \, A a c^{2} d x^{3} e^{6} + 432 \, A a c^{2} d^{2} x^{2} e^{5} + 108 \, A a c^{2} d^{3} x e^{4} + 12 \, A a c^{2} d^{4} e^{3} + 1260 \, B a^{2} c x^{3} e^{7} + 540 \, B a^{2} c d x^{2} e^{6} + 135 \, B a^{2} c d^{2} x e^{5} + 15 \, B a^{2} c d^{3} e^{4} + 1080 \, A a^{2} c x^{2} e^{7} + 270 \, A a^{2} c d x e^{6} + 30 \, A a^{2} c d^{2} e^{5} + 315 \, B a^{3} x e^{7} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{2520 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(1260*B*c^3*x^7*e^7 + 2940*B*c^3*d*x^6*e^6 + 4410*B*c^3*d^2*x^5*e^5 + 4410*B*c^3*d^3*x^4*e^4 + 2940*B*
c^3*d^4*x^3*e^3 + 1260*B*c^3*d^5*x^2*e^2 + 315*B*c^3*d^6*x*e + 35*B*c^3*d^7 + 840*A*c^3*x^6*e^7 + 1260*A*c^3*d
*x^5*e^6 + 1260*A*c^3*d^2*x^4*e^5 + 840*A*c^3*d^3*x^3*e^4 + 360*A*c^3*d^4*x^2*e^3 + 90*A*c^3*d^5*x*e^2 + 10*A*
c^3*d^6*e + 1890*B*a*c^2*x^5*e^7 + 1890*B*a*c^2*d*x^4*e^6 + 1260*B*a*c^2*d^2*x^3*e^5 + 540*B*a*c^2*d^3*x^2*e^4
+ 135*B*a*c^2*d^4*x*e^3 + 15*B*a*c^2*d^5*e^2 + 1512*A*a*c^2*x^4*e^7 + 1008*A*a*c^2*d*x^3*e^6 + 432*A*a*c^2*d^
2*x^2*e^5 + 108*A*a*c^2*d^3*x*e^4 + 12*A*a*c^2*d^4*e^3 + 1260*B*a^2*c*x^3*e^7 + 540*B*a^2*c*d*x^2*e^6 + 135*B*
a^2*c*d^2*x*e^5 + 15*B*a^2*c*d^3*e^4 + 1080*A*a^2*c*x^2*e^7 + 270*A*a^2*c*d*x*e^6 + 30*A*a^2*c*d^2*e^5 + 315*B
*a^3*x*e^7 + 35*B*a^3*d*e^6 + 280*A*a^3*e^7)*e^(-8)/(x*e + d)^9

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maple [A]  time = 0.05, size = 449, normalized size = 1.34 \begin {gather*} -\frac {B \,c^{3}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {\left (A e -7 B d \right ) c^{3}}{3 \left (e x +d \right )^{3} e^{8}}+\frac {3 \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right ) c^{2}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {\left (3 a A \,e^{3}+15 A c \,d^{2} e -15 a B d \,e^{2}-35 B c \,d^{3}\right ) c^{2}}{5 \left (e x +d \right )^{5} e^{8}}-\frac {3 \left (A \,a^{2} e^{5}+6 A \,d^{2} a c \,e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B \,d^{3} a c \,e^{2}-7 B \,c^{2} d^{5}\right ) c}{7 \left (e x +d \right )^{7} e^{8}}+\frac {\left (12 A d a c \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,a^{2} e^{4}-30 B \,d^{2} a c \,e^{2}-35 B \,c^{2} d^{4}\right ) c}{6 \left (e x +d \right )^{6} e^{8}}-\frac {A \,a^{3} e^{7}+3 A \,d^{2} a^{2} c \,e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,d^{6} c^{3} e -B d \,a^{3} e^{6}-3 B \,d^{3} a^{2} c \,e^{4}-3 B \,d^{5} a \,c^{2} e^{2}-B \,d^{7} c^{3}}{9 \left (e x +d \right )^{9} e^{8}}-\frac {-6 A d \,a^{2} c \,e^{5}-12 A \,d^{3} a \,c^{2} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,d^{2} a^{2} c \,e^{4}+15 B \,d^{4} a \,c^{2} e^{2}+7 B \,d^{6} c^{3}}{8 \left (e x +d \right )^{8} e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x)

[Out]

-1/9*(A*a^3*e^7+3*A*a^2*c*d^2*e^5+3*A*a*c^2*d^4*e^3+A*c^3*d^6*e-B*a^3*d*e^6-3*B*a^2*c*d^3*e^4-3*B*a*c^2*d^5*e^
2-B*c^3*d^7)/e^8/(e*x+d)^9+3/4*c^2*(2*A*c*d*e-B*a*e^2-7*B*c*d^2)/e^8/(e*x+d)^4-1/2*B*c^3/e^8/(e*x+d)^2-1/3*c^3
*(A*e-7*B*d)/e^8/(e*x+d)^3-1/8*(-6*A*a^2*c*d*e^5-12*A*a*c^2*d^3*e^3-6*A*c^3*d^5*e+B*a^3*e^6+9*B*a^2*c*d^2*e^4+
15*B*a*c^2*d^4*e^2+7*B*c^3*d^6)/e^8/(e*x+d)^8-3/7*c*(A*a^2*e^5+6*A*a*c*d^2*e^3+5*A*c^2*d^4*e-3*B*a^2*d*e^4-10*
B*a*c*d^3*e^2-7*B*c^2*d^5)/e^8/(e*x+d)^7+1/6*c*(12*A*a*c*d*e^3+20*A*c^2*d^3*e-3*B*a^2*e^4-30*B*a*c*d^2*e^2-35*
B*c^2*d^4)/e^8/(e*x+d)^6-1/5*c^2*(3*A*a*e^3+15*A*c*d^2*e-15*B*a*d*e^2-35*B*c*d^3)/e^8/(e*x+d)^5

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maxima [A]  time = 0.60, size = 546, normalized size = 1.63 \begin {gather*} -\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, A c^{3} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 12 \, A a c^{2} e^{7}\right )} x^{4} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} + 12 \, A a c^{2} d e^{6} + 15 \, B a^{2} c e^{7}\right )} x^{3} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} + 12 \, A a c^{2} d^{2} e^{5} + 15 \, B a^{2} c d e^{6} + 30 \, A a^{2} c e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} + 12 \, A a c^{2} d^{3} e^{4} + 15 \, B a^{2} c d^{2} e^{5} + 30 \, A a^{2} c d e^{6} + 35 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 10*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 + 12*A*a*c^2*d^4*e^3 + 15*B*a
^2*c*d^3*e^4 + 30*A*a^2*c*d^2*e^5 + 35*B*a^3*d*e^6 + 280*A*a^3*e^7 + 420*(7*B*c^3*d*e^6 + 2*A*c^3*e^7)*x^6 + 6
30*(7*B*c^3*d^2*e^5 + 2*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 126*(35*B*c^3*d^3*e^4 + 10*A*c^3*d^2*e^5 + 15*B*a*c
^2*d*e^6 + 12*A*a*c^2*e^7)*x^4 + 84*(35*B*c^3*d^4*e^3 + 10*A*c^3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 + 12*A*a*c^2*d*e
^6 + 15*B*a^2*c*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*A*c^3*d^4*e^3 + 15*B*a*c^2*d^3*e^4 + 12*A*a*c^2*d^2*e^5 +
15*B*a^2*c*d*e^6 + 30*A*a^2*c*e^7)*x^2 + 9*(35*B*c^3*d^6*e + 10*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 + 12*A*a*c
^2*d^3*e^4 + 15*B*a^2*c*d^2*e^5 + 30*A*a^2*c*d*e^6 + 35*B*a^3*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x
^7 + 84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x +
d^9*e^8)

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mupad [B]  time = 1.80, size = 513, normalized size = 1.54 \begin {gather*} -\frac {\frac {35\,B\,a^3\,d\,e^6+280\,A\,a^3\,e^7+15\,B\,a^2\,c\,d^3\,e^4+30\,A\,a^2\,c\,d^2\,e^5+15\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3+35\,B\,c^3\,d^7+10\,A\,c^3\,d^6\,e}{2520\,e^8}+\frac {x\,\left (35\,B\,a^3\,e^6+15\,B\,a^2\,c\,d^2\,e^4+30\,A\,a^2\,c\,d\,e^5+15\,B\,a\,c^2\,d^4\,e^2+12\,A\,a\,c^2\,d^3\,e^3+35\,B\,c^3\,d^6+10\,A\,c^3\,d^5\,e\right )}{280\,e^7}+\frac {c^2\,x^4\,\left (35\,B\,c\,d^3+10\,A\,c\,d^2\,e+15\,B\,a\,d\,e^2+12\,A\,a\,e^3\right )}{20\,e^4}+\frac {c\,x^3\,\left (15\,B\,a^2\,e^4+15\,B\,a\,c\,d^2\,e^2+12\,A\,a\,c\,d\,e^3+35\,B\,c^2\,d^4+10\,A\,c^2\,d^3\,e\right )}{30\,e^5}+\frac {c^3\,x^6\,\left (2\,A\,e+7\,B\,d\right )}{6\,e^2}+\frac {c^2\,x^5\,\left (7\,B\,c\,d^2+2\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{4\,e^3}+\frac {c\,x^2\,\left (15\,B\,a^2\,d\,e^4+30\,A\,a^2\,e^5+15\,B\,a\,c\,d^3\,e^2+12\,A\,a\,c\,d^2\,e^3+35\,B\,c^2\,d^5+10\,A\,c^2\,d^4\,e\right )}{70\,e^6}+\frac {B\,c^3\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^10,x)

[Out]

-((280*A*a^3*e^7 + 35*B*c^3*d^7 + 35*B*a^3*d*e^6 + 10*A*c^3*d^6*e + 12*A*a*c^2*d^4*e^3 + 30*A*a^2*c*d^2*e^5 +
15*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^4)/(2520*e^8) + (x*(35*B*a^3*e^6 + 35*B*c^3*d^6 + 10*A*c^3*d^5*e + 12*A*
a*c^2*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 15*B*a^2*c*d^2*e^4 + 30*A*a^2*c*d*e^5))/(280*e^7) + (c^2*x^4*(12*A*a*e^3
+ 35*B*c*d^3 + 15*B*a*d*e^2 + 10*A*c*d^2*e))/(20*e^4) + (c*x^3*(15*B*a^2*e^4 + 35*B*c^2*d^4 + 10*A*c^2*d^3*e +
12*A*a*c*d*e^3 + 15*B*a*c*d^2*e^2))/(30*e^5) + (c^3*x^6*(2*A*e + 7*B*d))/(6*e^2) + (c^2*x^5*(3*B*a*e^2 + 7*B*
c*d^2 + 2*A*c*d*e))/(4*e^3) + (c*x^2*(30*A*a^2*e^5 + 35*B*c^2*d^5 + 15*B*a^2*d*e^4 + 10*A*c^2*d^4*e + 12*A*a*c
*d^2*e^3 + 15*B*a*c*d^3*e^2))/(70*e^6) + (B*c^3*x^7)/(2*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84
*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**10,x)

[Out]

Timed out

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