3.12.50 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=313 \[ \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{2 e^8 (d+e x)^2}+\frac {3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}+\frac {c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^3}-\frac {c^3 x (6 B d-A e)}{e^7}+\frac {B c^3 x^2}{2 e^6} \]

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Rubi [A]  time = 0.34, antiderivative size = 313, 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 )}{2 e^8 (d+e x)^2}+\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 )}{e^8 (d+e x)}+\frac {3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac {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 )}{e^8 (d+e x)^3}-\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 )}{4 e^8 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}-\frac {c^3 x (6 B d-A e)}{e^7}+\frac {B c^3 x^2}{2 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

-((c^3*(6*B*d - A*e)*x)/e^7) + (B*c^3*x^2)/(2*e^6) + ((B*d - A*e)*(c*d^2 + a*e^2)^3)/(5*e^8*(d + e*x)^5) - ((c
*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(4*e^8*(d + e*x)^4) + (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))/(e^8*(d + e*x)^3) + (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)))/(2*e^8*(d + e*x)^2) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(e
^8*(d + e*x)) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*Log[d + e*x])/e^8

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

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Mathematica [A]  time = 0.20, size = 388, normalized size = 1.24 \begin {gather*} \frac {-2 A e \left (2 a^3 e^6+a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+B \left (-a^3 e^6 (d+5 e x)-3 a^2 c e^4 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+a c^2 d e^2 \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+c^3 \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )\right )+60 c^2 (d+e x)^5 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{20 e^8 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

(-2*A*e*(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 +
10*d*e^3*x^3 + 5*e^4*x^4) + c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 5
0*d*e^5*x^5 - 10*e^6*x^6)) + B*(-(a^3*e^6*(d + 5*e*x)) - 3*a^2*c*e^4*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*
x^3) + a*c^2*d*e^2*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + c^3*(459*d^7 + 1
875*d^6*e*x + 2700*d^5*e^2*x^2 + 1300*d^4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*
x^7)) + 60*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6, x]

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fricas [B]  time = 0.41, size = 730, normalized size = 2.33 \begin {gather*} \frac {10 \, B c^{3} e^{7} x^{7} + 459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} - 10 \, {\left (7 \, B c^{3} d e^{6} - 2 \, A c^{3} e^{7}\right )} x^{6} - 100 \, {\left (5 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} - 20 \, {\left (20 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, {\left (130 \, B c^{3} d^{4} e^{3} - 80 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (270 \, B c^{3} d^{5} e^{2} - 120 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (375 \, B c^{3} d^{6} e - 150 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x + 60 \, {\left (7 \, B c^{3} d^{7} - 2 \, A c^{3} d^{6} e + B a c^{2} d^{5} e^{2} + {\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 5 \, {\left (7 \, B c^{3} d^{3} e^{4} - 2 \, A c^{3} d^{2} e^{5} + B a c^{2} d e^{6}\right )} x^{4} + 10 \, {\left (7 \, B c^{3} d^{4} e^{3} - 2 \, A c^{3} d^{3} e^{4} + B a c^{2} d^{2} e^{5}\right )} x^{3} + 10 \, {\left (7 \, B c^{3} d^{5} e^{2} - 2 \, A c^{3} d^{4} e^{3} + B a c^{2} d^{3} e^{4}\right )} x^{2} + 5 \, {\left (7 \, B c^{3} d^{6} e - 2 \, A c^{3} d^{5} e^{2} + B a c^{2} d^{4} e^{3}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*
c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 - 10*(7*B*c^3*d*e^6 - 2*A*c^3*e^7)*x^6 - 100*(5*B*c^
3*d^2*e^5 - A*c^3*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4
+ 10*(130*B*c^3*d^4*e^3 - 80*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(
270*B*c^3*d^5*e^2 - 120*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c
*e^7)*x^2 + 5*(375*B*c^3*d^6*e - 150*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*
e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x + 60*(7*B*c^3*d^7 - 2*A*c^3*d^6*e + B*a*c^2*d^5*e^2 + (7*B*c^3*d^2*e^5 -
2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 5*(7*B*c^3*d^3*e^4 - 2*A*c^3*d^2*e^5 + B*a*c^2*d*e^6)*x^4 + 10*(7*B*c^3*d^4
*e^3 - 2*A*c^3*d^3*e^4 + B*a*c^2*d^2*e^5)*x^3 + 10*(7*B*c^3*d^5*e^2 - 2*A*c^3*d^4*e^3 + B*a*c^2*d^3*e^4)*x^2 +
5*(7*B*c^3*d^6*e - 2*A*c^3*d^5*e^2 + B*a*c^2*d^4*e^3)*x)*log(e*x + d))/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11
*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8)

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giac [A]  time = 0.16, size = 429, normalized size = 1.37 \begin {gather*} 3 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B c^{3} x^{2} e^{6} - 12 \, B c^{3} d x e^{5} + 2 \, A c^{3} x e^{6}\right )} e^{\left (-12\right )} + \frac {{\left (459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} + 20 \, {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} - 4 \, A a^{3} e^{7} + 10 \, {\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x\right )} e^{\left (-8\right )}}{20 \, {\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

3*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*e^(-8)*log(abs(x*e + d)) + 1/2*(B*c^3*x^2*e^6 - 12*B*c^3*d*x*e^5 +
2*A*c^3*x*e^6)*e^(-12) + 1/20*(459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3
*B*a^2*c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 + 20*(35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^
6 - 3*A*a*c^2*e^7)*x^4 - 4*A*a^3*e^7 + 10*(245*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a
*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2
*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3
- 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x)*e^(-8)/(x*e + d)^5

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maple [B]  time = 0.06, size = 646, normalized size = 2.06 \begin {gather*} -\frac {A \,a^{3}}{5 \left (e x +d \right )^{5} e}-\frac {3 A \,a^{2} c \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}-\frac {3 A a \,c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {A \,c^{3} d^{6}}{5 \left (e x +d \right )^{5} e^{7}}+\frac {B \,a^{3} d}{5 \left (e x +d \right )^{5} e^{2}}+\frac {3 B \,a^{2} c \,d^{3}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {3 B a \,c^{2} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}+\frac {B \,c^{3} d^{7}}{5 \left (e x +d \right )^{5} e^{8}}+\frac {3 A \,a^{2} c d}{2 \left (e x +d \right )^{4} e^{3}}+\frac {3 A a \,c^{2} d^{3}}{\left (e x +d \right )^{4} e^{5}}+\frac {3 A \,c^{3} d^{5}}{2 \left (e x +d \right )^{4} e^{7}}-\frac {B \,a^{3}}{4 \left (e x +d \right )^{4} e^{2}}-\frac {9 B \,a^{2} c \,d^{2}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {15 B a \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {7 B \,c^{3} d^{6}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {A \,a^{2} c}{\left (e x +d \right )^{3} e^{3}}-\frac {6 A a \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{5}}-\frac {5 A \,c^{3} d^{4}}{\left (e x +d \right )^{3} e^{7}}+\frac {3 B \,a^{2} c d}{\left (e x +d \right )^{3} e^{4}}+\frac {10 B a \,c^{2} d^{3}}{\left (e x +d \right )^{3} e^{6}}+\frac {7 B \,c^{3} d^{5}}{\left (e x +d \right )^{3} e^{8}}+\frac {6 A a \,c^{2} d}{\left (e x +d \right )^{2} e^{5}}+\frac {10 A \,c^{3} d^{3}}{\left (e x +d \right )^{2} e^{7}}-\frac {3 B \,a^{2} c}{2 \left (e x +d \right )^{2} e^{4}}-\frac {15 B a \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{6}}-\frac {35 B \,c^{3} d^{4}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {B \,c^{3} x^{2}}{2 e^{6}}-\frac {3 A a \,c^{2}}{\left (e x +d \right ) e^{5}}-\frac {15 A \,c^{3} d^{2}}{\left (e x +d \right ) e^{7}}-\frac {6 A \,c^{3} d \ln \left (e x +d \right )}{e^{7}}+\frac {A \,c^{3} x}{e^{6}}+\frac {15 B a \,c^{2} d}{\left (e x +d \right ) e^{6}}+\frac {3 B a \,c^{2} \ln \left (e x +d \right )}{e^{6}}+\frac {35 B \,c^{3} d^{3}}{\left (e x +d \right ) e^{8}}+\frac {21 B \,c^{3} d^{2} \ln \left (e x +d \right )}{e^{8}}-\frac {6 B \,c^{3} d x}{e^{7}} \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)^6,x)

[Out]

-1/5/e/(e*x+d)^5*A*a^3+c^3/e^6*A*x-1/4/e^2/(e*x+d)^4*B*a^3+15*c^2/e^6/(e*x+d)*a*B*d+3/2/e^3/(e*x+d)^4*A*a^2*c*
d-1/5/e^7/(e*x+d)^5*A*c^3*d^6+1/5/e^2/(e*x+d)^5*B*d*a^3-6*c^3/e^7*B*d*x+21*c^3/e^8*ln(e*x+d)*B*d^2-6*c^2/e^5/(
e*x+d)^3*A*d^2*a+3*c/e^4/(e*x+d)^3*B*a^2*d+10*c^2/e^6/(e*x+d)^3*B*d^3*a-3/5/e^3/(e*x+d)^5*A*d^2*a^2*c-3/5/e^5/
(e*x+d)^5*A*d^4*a*c^2+3/5/e^4/(e*x+d)^5*B*d^3*a^2*c+3/5/e^6/(e*x+d)^5*B*d^5*a*c^2-15*c^2/e^6/(e*x+d)^2*B*d^2*a
-15*c^3/e^7/(e*x+d)*A*d^2+10*c^3/e^7/(e*x+d)^2*A*d^3-3/2*c/e^4/(e*x+d)^2*B*a^2-35/2*c^3/e^8/(e*x+d)^2*B*d^4+1/
2*B*c^3*x^2/e^6+1/5/e^8/(e*x+d)^5*B*c^3*d^7-6*c^3/e^7*ln(e*x+d)*A*d+3*c^2/e^6*ln(e*x+d)*B*a-c/e^3/(e*x+d)^3*A*
a^2-5*c^3/e^7/(e*x+d)^3*A*d^4+7*c^3/e^8/(e*x+d)^3*B*d^5+35*c^3/e^8/(e*x+d)*B*d^3+3/2/e^7/(e*x+d)^4*A*c^3*d^5-7
/4/e^8/(e*x+d)^4*B*c^3*d^6-3*c^2/e^5/(e*x+d)*a*A+3/e^5/(e*x+d)^4*A*a*c^2*d^3-9/4/e^4/(e*x+d)^4*B*a^2*c*d^2-15/
4/e^6/(e*x+d)^4*B*a*c^2*d^4+6*c^2/e^5/(e*x+d)^2*A*d*a

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maxima [A]  time = 0.79, size = 499, normalized size = 1.59 \begin {gather*} \frac {459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} + 20 \, {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, {\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{20 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} + \frac {B c^{3} e x^{2} - 2 \, {\left (6 \, B c^{3} d - A c^{3} e\right )} x}{2 \, e^{7}} + \frac {3 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{8}} \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)^6,x, algorithm="maxima")

[Out]

1/20*(459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 2*A*a^2
*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 + 20*(35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^
2*e^7)*x^4 + 10*(245*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7
)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6
- 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*
B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2
+ 5*d^4*e^9*x + d^5*e^8) + 1/2*(B*c^3*e*x^2 - 2*(6*B*c^3*d - A*c^3*e)*x)/e^7 + 3*(7*B*c^3*d^2 - 2*A*c^3*d*e +
B*a*c^2*e^2)*log(e*x + d)/e^8

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mupad [B]  time = 0.16, size = 494, normalized size = 1.58 \begin {gather*} x\,\left (\frac {A\,c^3}{e^6}-\frac {6\,B\,c^3\,d}{e^7}\right )-\frac {\frac {B\,a^3\,d\,e^6+4\,A\,a^3\,e^7+3\,B\,a^2\,c\,d^3\,e^4+2\,A\,a^2\,c\,d^2\,e^5-137\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3-459\,B\,c^3\,d^7+174\,A\,c^3\,d^6\,e}{20\,e}+x^2\,\left (\frac {3\,B\,a^2\,c\,d\,e^5}{2}+A\,a^2\,c\,e^6-55\,B\,a\,c^2\,d^3\,e^3+6\,A\,a\,c^2\,d^2\,e^4-\frac {329\,B\,c^3\,d^5\,e}{2}+65\,A\,c^3\,d^4\,e^2\right )+x^3\,\left (\frac {3\,B\,a^2\,c\,e^6}{2}-45\,B\,a\,c^2\,d^2\,e^4+6\,A\,a\,c^2\,d\,e^5-\frac {245\,B\,c^3\,d^4\,e^2}{2}+50\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{4}+\frac {3\,B\,a^2\,c\,d^2\,e^4}{4}+\frac {A\,a^2\,c\,d\,e^5}{2}-\frac {125\,B\,a\,c^2\,d^4\,e^2}{4}+3\,A\,a\,c^2\,d^3\,e^3-\frac {399\,B\,c^3\,d^6}{4}+\frac {77\,A\,c^3\,d^5\,e}{2}\right )+x^4\,\left (-35\,B\,c^3\,d^3\,e^3+15\,A\,c^3\,d^2\,e^4-15\,B\,a\,c^2\,d\,e^5+3\,A\,a\,c^2\,e^6\right )}{d^5\,e^7+5\,d^4\,e^8\,x+10\,d^3\,e^9\,x^2+10\,d^2\,e^{10}\,x^3+5\,d\,e^{11}\,x^4+e^{12}\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (21\,B\,c^3\,d^2-6\,A\,c^3\,d\,e+3\,B\,a\,c^2\,e^2\right )}{e^8}+\frac {B\,c^3\,x^2}{2\,e^6} \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)^6,x)

[Out]

x*((A*c^3)/e^6 - (6*B*c^3*d)/e^7) - ((4*A*a^3*e^7 - 459*B*c^3*d^7 + B*a^3*d*e^6 + 174*A*c^3*d^6*e + 12*A*a*c^2
*d^4*e^3 + 2*A*a^2*c*d^2*e^5 - 137*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4)/(20*e) + x^2*(A*a^2*c*e^6 - (329*B*c^3
*d^5*e)/2 + 65*A*c^3*d^4*e^2 + 6*A*a*c^2*d^2*e^4 - 55*B*a*c^2*d^3*e^3 + (3*B*a^2*c*d*e^5)/2) + x^3*((3*B*a^2*c
*e^6)/2 + 50*A*c^3*d^3*e^3 - (245*B*c^3*d^4*e^2)/2 - 45*B*a*c^2*d^2*e^4 + 6*A*a*c^2*d*e^5) + x*((B*a^3*e^6)/4
- (399*B*c^3*d^6)/4 + (77*A*c^3*d^5*e)/2 + 3*A*a*c^2*d^3*e^3 - (125*B*a*c^2*d^4*e^2)/4 + (3*B*a^2*c*d^2*e^4)/4
+ (A*a^2*c*d*e^5)/2) + x^4*(3*A*a*c^2*e^6 + 15*A*c^3*d^2*e^4 - 35*B*c^3*d^3*e^3 - 15*B*a*c^2*d*e^5))/(d^5*e^7
+ e^12*x^5 + 5*d^4*e^8*x + 5*d*e^11*x^4 + 10*d^3*e^9*x^2 + 10*d^2*e^10*x^3) + (log(d + e*x)*(21*B*c^3*d^2 - 6
*A*c^3*d*e + 3*B*a*c^2*e^2))/e^8 + (B*c^3*x^2)/(2*e^6)

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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)**6,x)

[Out]

Timed out

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