3.1321 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=300 \[ -\frac {c x \left (A c d e \left (9 a e^2+10 c d^2\right )-3 B \left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}+\frac {c^2 x^3 \left (a B e^2-A c d e+2 B c d^2\right )}{e^5}-\frac {c^2 x^2 \left (-3 a A e^3+9 a B d e^2-6 A c d^2 e+10 B c d^3\right )}{2 e^6}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {3 c \left (a e^2+c d^2\right ) \log (d+e x) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac {c^3 x^4 (3 B d-A e)}{4 e^4}+\frac {B c^3 x^5}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.42, antiderivative size = 300, 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} \[ -\frac {c x \left (A c d e \left (9 a e^2+10 c d^2\right )-3 B \left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}+\frac {c^2 x^3 \left (a B e^2-A c d e+2 B c d^2\right )}{e^5}-\frac {c^2 x^2 \left (-3 a A e^3+9 a B d e^2-6 A c d^2 e+10 B c d^3\right )}{2 e^6}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {3 c \left (a e^2+c d^2\right ) \log (d+e x) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac {c^3 x^4 (3 B d-A e)}{4 e^4}+\frac {B c^3 x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.19, size = 414, normalized size = 1.38 \[ \frac {5 A e \left (-2 a^3 e^6+6 a^2 c d e^4 (3 d+4 e x)+6 a c^2 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )\right )+B \left (-10 a^3 e^6 (d+2 e x)+30 a^2 c e^4 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+10 a c^2 e^2 \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+c^3 \left (-130 d^7+160 d^6 e x+500 d^5 e^2 x^2+140 d^4 e^3 x^3-35 d^3 e^4 x^4+14 d^2 e^5 x^5-7 d e^6 x^6+4 e^7 x^7\right )\right )-60 c (d+e x)^2 \left (a e^2+c d^2\right ) \log (d+e x) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{20 e^8 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*A*e*(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 4*e*x) + 6*a*c^2*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x
^3 + e^4*x^4) + c^3*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6
*x^6)) + B*(-10*a^3*e^6*(d + 2*e*x) + 30*a^2*c*e^4*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 10*a*c^2*e
^2*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + c^3*(-130*d^7 + 160*d^6
*e*x + 500*d^5*e^2*x^2 + 140*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7*d*e^6*x^6 + 4*e^7*x^7)) - 60*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)*(d + e*x)^2*Log[d + e*x])/(20*e^8*(d + e*x)^
2)

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fricas [B]  time = 0.64, size = 691, normalized size = 2.30 \[ \frac {4 \, B c^{3} e^{7} x^{7} - 130 \, B c^{3} d^{7} + 110 \, A c^{3} d^{6} e - 270 \, B a c^{2} d^{5} e^{2} + 210 \, A a c^{2} d^{4} e^{3} - 150 \, B a^{2} c d^{3} e^{4} + 90 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 10 \, A a^{3} e^{7} - {\left (7 \, B c^{3} d e^{6} - 5 \, A c^{3} e^{7}\right )} x^{6} + 2 \, {\left (7 \, B c^{3} d^{2} e^{5} - 5 \, A c^{3} d e^{6} + 10 \, B a c^{2} e^{7}\right )} x^{5} - 5 \, {\left (7 \, B c^{3} d^{3} e^{4} - 5 \, A c^{3} d^{2} e^{5} + 10 \, B a c^{2} d e^{6} - 6 \, A a c^{2} e^{7}\right )} x^{4} + 20 \, {\left (7 \, B c^{3} d^{4} e^{3} - 5 \, A c^{3} d^{3} e^{4} + 10 \, B a c^{2} d^{2} e^{5} - 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (50 \, B c^{3} d^{5} e^{2} - 34 \, A c^{3} d^{4} e^{3} + 63 \, B a c^{2} d^{3} e^{4} - 33 \, A a c^{2} d^{2} e^{5} + 12 \, B a^{2} c d e^{6}\right )} x^{2} + 20 \, {\left (8 \, B c^{3} d^{6} e - 4 \, A c^{3} d^{5} e^{2} + 3 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} - 6 \, B a^{2} c d^{2} e^{5} + 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x - 60 \, {\left (7 \, B c^{3} d^{7} - 5 \, A c^{3} d^{6} e + 10 \, B a c^{2} d^{5} e^{2} - 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - A a^{2} c d^{2} e^{5} + {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B a c^{2} d^{3} e^{4} - 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - A a^{2} c e^{7}\right )} x^{2} + 2 \, {\left (7 \, B c^{3} d^{6} e - 5 \, A c^{3} d^{5} e^{2} + 10 \, B a c^{2} d^{4} e^{3} - 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} - A a^{2} c d e^{6}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*B*c^3*e^7*x^7 - 130*B*c^3*d^7 + 110*A*c^3*d^6*e - 270*B*a*c^2*d^5*e^2 + 210*A*a*c^2*d^4*e^3 - 150*B*a^
2*c*d^3*e^4 + 90*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 10*A*a^3*e^7 - (7*B*c^3*d*e^6 - 5*A*c^3*e^7)*x^6 + 2*(7*B*
c^3*d^2*e^5 - 5*A*c^3*d*e^6 + 10*B*a*c^2*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 5*A*c^3*d^2*e^5 + 10*B*a*c^2*d*e^6 -
6*A*a*c^2*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 - 5*A*c^3*d^3*e^4 + 10*B*a*c^2*d^2*e^5 - 6*A*a*c^2*d*e^6 + 3*B*a^2*c*
e^7)*x^3 + 10*(50*B*c^3*d^5*e^2 - 34*A*c^3*d^4*e^3 + 63*B*a*c^2*d^3*e^4 - 33*A*a*c^2*d^2*e^5 + 12*B*a^2*c*d*e^
6)*x^2 + 20*(8*B*c^3*d^6*e - 4*A*c^3*d^5*e^2 + 3*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4 - 6*B*a^2*c*d^2*e^5 + 6*A
*a^2*c*d*e^6 - B*a^3*e^7)*x - 60*(7*B*c^3*d^7 - 5*A*c^3*d^6*e + 10*B*a*c^2*d^5*e^2 - 6*A*a*c^2*d^4*e^3 + 3*B*a
^2*c*d^3*e^4 - A*a^2*c*d^2*e^5 + (7*B*c^3*d^5*e^2 - 5*A*c^3*d^4*e^3 + 10*B*a*c^2*d^3*e^4 - 6*A*a*c^2*d^2*e^5 +
3*B*a^2*c*d*e^6 - A*a^2*c*e^7)*x^2 + 2*(7*B*c^3*d^6*e - 5*A*c^3*d^5*e^2 + 10*B*a*c^2*d^4*e^3 - 6*A*a*c^2*d^3*
e^4 + 3*B*a^2*c*d^2*e^5 - A*a^2*c*d*e^6)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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giac [A]  time = 0.16, size = 440, normalized size = 1.47 \[ -3 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{20} \, {\left (4 \, B c^{3} x^{5} e^{12} - 15 \, B c^{3} d x^{4} e^{11} + 40 \, B c^{3} d^{2} x^{3} e^{10} - 100 \, B c^{3} d^{3} x^{2} e^{9} + 300 \, B c^{3} d^{4} x e^{8} + 5 \, A c^{3} x^{4} e^{12} - 20 \, A c^{3} d x^{3} e^{11} + 60 \, A c^{3} d^{2} x^{2} e^{10} - 200 \, A c^{3} d^{3} x e^{9} + 20 \, B a c^{2} x^{3} e^{12} - 90 \, B a c^{2} d x^{2} e^{11} + 360 \, B a c^{2} d^{2} x e^{10} + 30 \, A a c^{2} x^{2} e^{12} - 180 \, A a c^{2} d x e^{11} + 60 \, B a^{2} c x e^{12}\right )} e^{\left (-15\right )} - \frac {{\left (13 \, B c^{3} d^{7} - 11 \, A c^{3} d^{6} e + 27 \, B a c^{2} d^{5} e^{2} - 21 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + A a^{3} e^{7} + 2 \, {\left (7 \, B c^{3} d^{6} e - 6 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x\right )} e^{\left (-8\right )}}{2 \, {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c*d*e^4 - A*a^2*c*e^5)*e^(-
8)*log(abs(x*e + d)) + 1/20*(4*B*c^3*x^5*e^12 - 15*B*c^3*d*x^4*e^11 + 40*B*c^3*d^2*x^3*e^10 - 100*B*c^3*d^3*x^
2*e^9 + 300*B*c^3*d^4*x*e^8 + 5*A*c^3*x^4*e^12 - 20*A*c^3*d*x^3*e^11 + 60*A*c^3*d^2*x^2*e^10 - 200*A*c^3*d^3*x
*e^9 + 20*B*a*c^2*x^3*e^12 - 90*B*a*c^2*d*x^2*e^11 + 360*B*a*c^2*d^2*x*e^10 + 30*A*a*c^2*x^2*e^12 - 180*A*a*c^
2*d*x*e^11 + 60*B*a^2*c*x*e^12)*e^(-15) - 1/2*(13*B*c^3*d^7 - 11*A*c^3*d^6*e + 27*B*a*c^2*d^5*e^2 - 21*A*a*c^2
*d^4*e^3 + 15*B*a^2*c*d^3*e^4 - 9*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 + A*a^3*e^7 + 2*(7*B*c^3*d^6*e - 6*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 + 9*B*a^2*c*d^2*e^5 - 6*A*a^2*c*d*e^6 + B*a^3*e^7)*x)*e^(-8)/(x*e
+ d)^2

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maple [B]  time = 0.06, size = 589, normalized size = 1.96 \[ \frac {B \,c^{3} x^{5}}{5 e^{3}}+\frac {A \,c^{3} x^{4}}{4 e^{3}}-\frac {3 B \,c^{3} d \,x^{4}}{4 e^{4}}-\frac {A \,c^{3} d \,x^{3}}{e^{4}}+\frac {B a \,c^{2} x^{3}}{e^{3}}+\frac {2 B \,c^{3} d^{2} x^{3}}{e^{5}}-\frac {A \,a^{3}}{2 \left (e x +d \right )^{2} e}-\frac {3 A \,a^{2} c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}-\frac {3 A a \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {3 A a \,c^{2} x^{2}}{2 e^{3}}-\frac {A \,c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {3 A \,c^{3} d^{2} x^{2}}{e^{5}}+\frac {B \,a^{3} d}{2 \left (e x +d \right )^{2} e^{2}}+\frac {3 B \,a^{2} c \,d^{3}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {3 B a \,c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {9 B a \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {B \,c^{3} d^{7}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {5 B \,c^{3} d^{3} x^{2}}{e^{6}}+\frac {6 A \,a^{2} c d}{\left (e x +d \right ) e^{3}}+\frac {3 A \,a^{2} c \ln \left (e x +d \right )}{e^{3}}+\frac {12 A a \,c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {18 A a \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {9 A a \,c^{2} d x}{e^{4}}+\frac {6 A \,c^{3} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {15 A \,c^{3} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {10 A \,c^{3} d^{3} x}{e^{6}}-\frac {B \,a^{3}}{\left (e x +d \right ) e^{2}}-\frac {9 B \,a^{2} c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {9 B \,a^{2} c d \ln \left (e x +d \right )}{e^{4}}+\frac {3 B \,a^{2} c x}{e^{3}}-\frac {15 B a \,c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {30 B a \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {18 B a \,c^{2} d^{2} x}{e^{5}}-\frac {7 B \,c^{3} d^{6}}{\left (e x +d \right ) e^{8}}-\frac {21 B \,c^{3} d^{5} \ln \left (e x +d \right )}{e^{8}}+\frac {15 B \,c^{3} d^{4} x}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3/e^4*A*x^3*d-7/e^8/(e*x+d)*B*c^3*d^6-1/2/e^7/(e*x+d)^2*A*c^3*d^6+1/2/e^2/(e*x+d)^2*B*d*a^3+6/e^7/(e*x+d)*A
*c^3*d^5+1/4*c^3/e^3*A*x^4-1/e^2/(e*x+d)*B*a^3-1/2/e/(e*x+d)^2*A*a^3-15/e^6/(e*x+d)*B*a*c^2*d^4-3/2/e^3/(e*x+d
)^2*A*d^2*a^2*c-3/2/e^5/(e*x+d)^2*A*a*c^2*d^4+3/2/e^4/(e*x+d)^2*B*a^2*c*d^3+3/2/e^6/(e*x+d)^2*B*a*c^2*d^5+3*c^
3/e^5*A*x^2*d^2+1/2/e^8/(e*x+d)^2*B*c^3*d^7+3*c/e^3*ln(e*x+d)*A*a^2+15*c^3/e^7*ln(e*x+d)*A*d^4-21*c^3/e^8*ln(e
*x+d)*B*d^5+15*c^3/e^7*B*x*d^4-3/4*c^3/e^4*B*x^4*d+18*c^2/e^5*ln(e*x+d)*A*d^2*a-9*c/e^4*ln(e*x+d)*B*d*a^2-30*c
^2/e^6*ln(e*x+d)*B*d^3*a-9/2*c^2/e^4*B*x^2*a*d-9*c^2/e^4*A*x*a*d+18*c^2/e^5*B*x*a*d^2+6/e^3/(e*x+d)*A*a^2*c*d+
12/e^5/(e*x+d)*A*a*c^2*d^3-9/e^4/(e*x+d)*B*a^2*c*d^2+3/2*c^2/e^3*A*x^2*a-5*c^3/e^6*B*x^2*d^3-10*c^3/e^6*A*x*d^
3+3*c/e^3*B*x*a^2+c^2/e^3*B*x^3*a+2*c^3/e^5*B*x^3*d^2+1/5*B*c^3/e^3*x^5

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maxima [A]  time = 0.69, size = 464, normalized size = 1.55 \[ -\frac {13 \, B c^{3} d^{7} - 11 \, A c^{3} d^{6} e + 27 \, B a c^{2} d^{5} e^{2} - 21 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + A a^{3} e^{7} + 2 \, {\left (7 \, B c^{3} d^{6} e - 6 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B c^{3} e^{4} x^{5} - 5 \, {\left (3 \, B c^{3} d e^{3} - A c^{3} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B c^{3} d^{2} e^{2} - A c^{3} d e^{3} + B a c^{2} e^{4}\right )} x^{3} - 10 \, {\left (10 \, B c^{3} d^{3} e - 6 \, A c^{3} d^{2} e^{2} + 9 \, B a c^{2} d e^{3} - 3 \, A a c^{2} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B c^{3} d^{4} - 10 \, A c^{3} d^{3} e + 18 \, B a c^{2} d^{2} e^{2} - 9 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(13*B*c^3*d^7 - 11*A*c^3*d^6*e + 27*B*a*c^2*d^5*e^2 - 21*A*a*c^2*d^4*e^3 + 15*B*a^2*c*d^3*e^4 - 9*A*a^2*c
*d^2*e^5 + B*a^3*d*e^6 + A*a^3*e^7 + 2*(7*B*c^3*d^6*e - 6*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 + 9*B*a^2*c*d^2*e^5 - 6*A*a^2*c*d*e^6 + B*a^3*e^7)*x)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*B*c^3*e^4
*x^5 - 5*(3*B*c^3*d*e^3 - A*c^3*e^4)*x^4 + 20*(2*B*c^3*d^2*e^2 - A*c^3*d*e^3 + B*a*c^2*e^4)*x^3 - 10*(10*B*c^3
*d^3*e - 6*A*c^3*d^2*e^2 + 9*B*a*c^2*d*e^3 - 3*A*a*c^2*e^4)*x^2 + 20*(15*B*c^3*d^4 - 10*A*c^3*d^3*e + 18*B*a*c
^2*d^2*e^2 - 9*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*x)/e^7 - 3*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6
*A*a*c^2*d^2*e^3 + 3*B*a^2*c*d*e^4 - A*a^2*c*e^5)*log(e*x + d)/e^8

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mupad [B]  time = 0.14, size = 681, normalized size = 2.27 \[ x^4\,\left (\frac {A\,c^3}{4\,e^3}-\frac {3\,B\,c^3\,d}{4\,e^4}\right )-\frac {\frac {B\,a^3\,d\,e^6+A\,a^3\,e^7+15\,B\,a^2\,c\,d^3\,e^4-9\,A\,a^2\,c\,d^2\,e^5+27\,B\,a\,c^2\,d^5\,e^2-21\,A\,a\,c^2\,d^4\,e^3+13\,B\,c^3\,d^7-11\,A\,c^3\,d^6\,e}{2\,e}+x\,\left (B\,a^3\,e^6+9\,B\,a^2\,c\,d^2\,e^4-6\,A\,a^2\,c\,d\,e^5+15\,B\,a\,c^2\,d^4\,e^2-12\,A\,a\,c^2\,d^3\,e^3+7\,B\,c^3\,d^6-6\,A\,c^3\,d^5\,e\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}-x\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^3}-\frac {B\,c^3\,d^3}{e^6}\right )}{e}-\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e^2}+\frac {d^3\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^3}-\frac {3\,B\,a^2\,c}{e^3}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {B\,a\,c^2}{e^3}+\frac {B\,c^3\,d^2}{e^5}\right )+x^2\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{2\,e}-\frac {3\,d^2\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{2\,e^2}+\frac {3\,A\,a\,c^2}{2\,e^3}-\frac {B\,c^3\,d^3}{2\,e^6}\right )-\frac {\ln \left (d+e\,x\right )\,\left (9\,B\,a^2\,c\,d\,e^4-3\,A\,a^2\,c\,e^5+30\,B\,a\,c^2\,d^3\,e^2-18\,A\,a\,c^2\,d^2\,e^3+21\,B\,c^3\,d^5-15\,A\,c^3\,d^4\,e\right )}{e^8}+\frac {B\,c^3\,x^5}{5\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((A*c^3)/(4*e^3) - (3*B*c^3*d)/(4*e^4)) - ((A*a^3*e^7 + 13*B*c^3*d^7 + B*a^3*d*e^6 - 11*A*c^3*d^6*e - 21*A
*a*c^2*d^4*e^3 - 9*A*a^2*c*d^2*e^5 + 27*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^4)/(2*e) + x*(B*a^3*e^6 + 7*B*c^3*d
^6 - 6*A*c^3*d^5*e - 12*A*a*c^2*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5))/(d^2*e^7
+ e^9*x^2 + 2*d*e^8*x) - x*((3*d*((3*d*((3*d*((A*c^3)/e^3 - (3*B*c^3*d)/e^4))/e - (3*B*a*c^2)/e^3 + (3*B*c^3*d
^2)/e^5))/e - (3*d^2*((A*c^3)/e^3 - (3*B*c^3*d)/e^4))/e^2 + (3*A*a*c^2)/e^3 - (B*c^3*d^3)/e^6))/e - (3*d^2*((3
*d*((A*c^3)/e^3 - (3*B*c^3*d)/e^4))/e - (3*B*a*c^2)/e^3 + (3*B*c^3*d^2)/e^5))/e^2 + (d^3*((A*c^3)/e^3 - (3*B*c
^3*d)/e^4))/e^3 - (3*B*a^2*c)/e^3) - x^3*((d*((A*c^3)/e^3 - (3*B*c^3*d)/e^4))/e - (B*a*c^2)/e^3 + (B*c^3*d^2)/
e^5) + x^2*((3*d*((3*d*((A*c^3)/e^3 - (3*B*c^3*d)/e^4))/e - (3*B*a*c^2)/e^3 + (3*B*c^3*d^2)/e^5))/(2*e) - (3*d
^2*((A*c^3)/e^3 - (3*B*c^3*d)/e^4))/(2*e^2) + (3*A*a*c^2)/(2*e^3) - (B*c^3*d^3)/(2*e^6)) - (log(d + e*x)*(21*B
*c^3*d^5 - 3*A*a^2*c*e^5 - 15*A*c^3*d^4*e - 18*A*a*c^2*d^2*e^3 + 30*B*a*c^2*d^3*e^2 + 9*B*a^2*c*d*e^4))/e^8 +
(B*c^3*x^5)/(5*e^3)

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sympy [A]  time = 6.10, size = 490, normalized size = 1.63 \[ \frac {B c^{3} x^{5}}{5 e^{3}} - \frac {3 c \left (a e^{2} + c d^{2}\right ) \left (- A a e^{3} - 5 A c d^{2} e + 3 B a d e^{2} + 7 B c d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A c^{3}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}}\right ) + x^{3} \left (- \frac {A c^{3} d}{e^{4}} + \frac {B a c^{2}}{e^{3}} + \frac {2 B c^{3} d^{2}}{e^{5}}\right ) + x^{2} \left (\frac {3 A a c^{2}}{2 e^{3}} + \frac {3 A c^{3} d^{2}}{e^{5}} - \frac {9 B a c^{2} d}{2 e^{4}} - \frac {5 B c^{3} d^{3}}{e^{6}}\right ) + x \left (- \frac {9 A a c^{2} d}{e^{4}} - \frac {10 A c^{3} d^{3}}{e^{6}} + \frac {3 B a^{2} c}{e^{3}} + \frac {18 B a c^{2} d^{2}}{e^{5}} + \frac {15 B c^{3} d^{4}}{e^{7}}\right ) + \frac {- A a^{3} e^{7} + 9 A a^{2} c d^{2} e^{5} + 21 A a c^{2} d^{4} e^{3} + 11 A c^{3} d^{6} e - B a^{3} d e^{6} - 15 B a^{2} c d^{3} e^{4} - 27 B a c^{2} d^{5} e^{2} - 13 B c^{3} d^{7} + x \left (12 A a^{2} c d e^{6} + 24 A a c^{2} d^{3} e^{4} + 12 A c^{3} d^{5} e^{2} - 2 B a^{3} e^{7} - 18 B a^{2} c d^{2} e^{5} - 30 B a c^{2} d^{4} e^{3} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*c**3*x**5/(5*e**3) - 3*c*(a*e**2 + c*d**2)*(-A*a*e**3 - 5*A*c*d**2*e + 3*B*a*d*e**2 + 7*B*c*d**3)*log(d + e*
x)/e**8 + x**4*(A*c**3/(4*e**3) - 3*B*c**3*d/(4*e**4)) + x**3*(-A*c**3*d/e**4 + B*a*c**2/e**3 + 2*B*c**3*d**2/
e**5) + x**2*(3*A*a*c**2/(2*e**3) + 3*A*c**3*d**2/e**5 - 9*B*a*c**2*d/(2*e**4) - 5*B*c**3*d**3/e**6) + x*(-9*A
*a*c**2*d/e**4 - 10*A*c**3*d**3/e**6 + 3*B*a**2*c/e**3 + 18*B*a*c**2*d**2/e**5 + 15*B*c**3*d**4/e**7) + (-A*a*
*3*e**7 + 9*A*a**2*c*d**2*e**5 + 21*A*a*c**2*d**4*e**3 + 11*A*c**3*d**6*e - B*a**3*d*e**6 - 15*B*a**2*c*d**3*e
**4 - 27*B*a*c**2*d**5*e**2 - 13*B*c**3*d**7 + x*(12*A*a**2*c*d*e**6 + 24*A*a*c**2*d**3*e**4 + 12*A*c**3*d**5*
e**2 - 2*B*a**3*e**7 - 18*B*a**2*c*d**2*e**5 - 30*B*a*c**2*d**4*e**3 - 14*B*c**3*d**6*e))/(2*d**2*e**8 + 4*d*e
**9*x + 2*e**10*x**2)

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