3.1325 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=320 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^3}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{2 e^8 (d+e x)^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8 (d+e x)^6}-\frac{c^3 (7 B d-A e) \log (d+e x)}{e^8}+\frac{B c^3 x}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.319566, antiderivative size = 320, 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, Rules used = {772} \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^3}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{2 e^8 (d+e x)^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8 (d+e x)^6}-\frac{c^3 (7 B d-A e) \log (d+e x)}{e^8}+\frac{B c^3 x}{e^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.225845, size = 377, normalized size = 1.18 \[ -\frac{A e \left (3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )+10 a^3 e^6+6 a c^2 e^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )-c^3 d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )\right )+B \left (3 a^2 c e^4 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 a^3 e^6 (d+6 e x)+30 a c^2 e^2 \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )+c^3 \left (7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5+3594 d^6 e x+669 d^7-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 c^3 (d+e x)^6 (7 B d-A e) \log (d+e x)}{60 e^8 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^7,x]

[Out]

-(A*e*(10*a^3*e^6 + 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 +
20*d*e^3*x^3 + 15*e^4*x^4) - c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*
x^4 + 360*e^5*x^5)) + B*(2*a^3*e^6*(d + 6*e*x) + 3*a^2*c*e^4*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 3
0*a*c^2*e^2*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) + c^3*(669*d^7 + 35
94*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7
*x^7)) + 60*c^3*(7*B*d - A*e)*(d + e*x)^6*Log[d + e*x])/(60*e^8*(d + e*x)^6)

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Maple [B]  time = 0.015, size = 656, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^7,x)

[Out]

B*c^3*x/e^7-3/e^6/(e*x+d)^5*B*a*c^2*d^4-9/2*c^2/e^5/(e*x+d)^4*A*d^2*a+9/4*c/e^4/(e*x+d)^4*B*a^2*d+15/2*c^2/e^6
/(e*x+d)^4*a*B*d^3-10*c^2/e^6/(e*x+d)^3*B*a*d^2+15/2*c^2/e^6/(e*x+d)^2*a*B*d+4*c^2/e^5/(e*x+d)^3*A*d*a-9/5/e^4
/(e*x+d)^5*B*a^2*c*d^2+1/2/e^6/(e*x+d)^6*B*d^5*a*c^2+6/5/e^3/(e*x+d)^5*A*a^2*c*d+12/5/e^5/(e*x+d)^5*A*a*c^2*d^
3+1/2/e^4/(e*x+d)^6*B*d^3*a^2*c-1/2/e^5/(e*x+d)^6*A*d^4*a*c^2-1/2/e^3/(e*x+d)^6*A*d^2*a^2*c-1/6/e/(e*x+d)^6*A*
a^3+c^3/e^7*ln(e*x+d)*A-1/5/e^2/(e*x+d)^5*B*a^3+35/2*c^3/e^8/(e*x+d)^2*B*d^3+6*c^3/e^7/(e*x+d)*A*d-3*c^2/e^6/(
e*x+d)*a*B-21*c^3/e^8/(e*x+d)*B*d^2-3/4*c/e^3/(e*x+d)^4*A*a^2-15/4*c^3/e^7/(e*x+d)^4*A*d^4+21/4*c^3/e^8/(e*x+d
)^4*B*d^5-7*c^3/e^8*ln(e*x+d)*B*d+20/3*c^3/e^7/(e*x+d)^3*A*d^3-c/e^4/(e*x+d)^3*B*a^2-35/3*c^3/e^8/(e*x+d)^3*B*
d^4-3/2*c^2/e^5/(e*x+d)^2*a*A-15/2*c^3/e^7/(e*x+d)^2*A*d^2-1/6/e^7/(e*x+d)^6*A*d^6*c^3+1/6/e^2/(e*x+d)^6*B*d*a
^3+1/6/e^8/(e*x+d)^6*B*c^3*d^7+6/5/e^7/(e*x+d)^5*A*c^3*d^5-7/5/e^8/(e*x+d)^5*B*c^3*d^6

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Maxima [A]  time = 1.19708, size = 690, normalized size = 2.16 \begin{align*} -\frac{669 \, B c^{3} d^{7} - 147 \, A c^{3} d^{6} e + 30 \, 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} + 3 \, A a^{2} c d^{2} e^{5} + 2 \, B a^{3} d e^{6} + 10 \, A a^{3} e^{7} + 180 \,{\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} + 30 \,{\left (175 \, B c^{3} d^{3} e^{4} - 45 \, 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} + 20 \,{\left (455 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, 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} + 15 \,{\left (539 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, 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} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \,{\left (609 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, 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} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x}{60 \,{\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} + \frac{B c^{3} x}{e^{7}} - \frac{{\left (7 \, B c^{3} d - A c^{3} e\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/60*(669*B*c^3*d^7 - 147*A*c^3*d^6*e + 30*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 + 3*A*a^2*
c*d^2*e^5 + 2*B*a^3*d*e^6 + 10*A*a^3*e^7 + 180*(7*B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 30*(175*B
*c^3*d^3*e^4 - 45*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 20*(455*B*c^3*d^4*e^3 - 110*A*c^3*d^
3*e^4 + 30*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 + 15*(539*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3
+ 30*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 + 3*A*a^2*c*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 137*A*c
^3*d^5*e^2 + 30*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 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x)/(e
^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) + B*c^3*
x/e^7 - (7*B*c^3*d - A*c^3*e)*log(e*x + d)/e^8

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Fricas [B]  time = 1.77097, size = 1466, normalized size = 4.58 \begin{align*} \frac{60 \, B c^{3} e^{7} x^{7} + 360 \, B c^{3} d e^{6} x^{6} - 669 \, B c^{3} d^{7} + 147 \, A c^{3} d^{6} e - 30 \, 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} - 3 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 10 \, A a^{3} e^{7} - 180 \,{\left (2 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} - 90 \,{\left (45 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + A a c^{2} e^{7}\right )} x^{4} - 20 \,{\left (410 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, 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} - 15 \,{\left (515 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, 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} + 3 \, A a^{2} c e^{7}\right )} x^{2} - 6 \,{\left (599 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, 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} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x - 60 \,{\left (7 \, B c^{3} d^{7} - A c^{3} d^{6} e +{\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 6 \,{\left (7 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} + 15 \,{\left (7 \, B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5}\right )} x^{4} + 20 \,{\left (7 \, B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4}\right )} x^{3} + 15 \,{\left (7 \, B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3}\right )} x^{2} + 6 \,{\left (7 \, B c^{3} d^{6} e - A c^{3} d^{5} e^{2}\right )} x\right )} \log \left (e x + d\right )}{60 \,{\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/60*(60*B*c^3*e^7*x^7 + 360*B*c^3*d*e^6*x^6 - 669*B*c^3*d^7 + 147*A*c^3*d^6*e - 30*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 - 3*A*a^2*c*d^2*e^5 - 2*B*a^3*d*e^6 - 10*A*a^3*e^7 - 180*(2*B*c^3*d^2*e^5 - 2*A*
c^3*d*e^6 + B*a*c^2*e^7)*x^5 - 90*(45*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 5*B*a*c^2*d*e^6 + A*a*c^2*e^7)*x^4 -
20*(410*B*c^3*d^4*e^3 - 110*A*c^3*d^3*e^4 + 30*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 - 15*(51
5*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30*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 + 3*A*a^2*c*e^7
)*x^2 - 6*(599*B*c^3*d^6*e - 137*A*c^3*d^5*e^2 + 30*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 +
3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x - 60*(7*B*c^3*d^7 - A*c^3*d^6*e + (7*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 6*(7*B*c^
3*d^2*e^5 - A*c^3*d*e^6)*x^5 + 15*(7*B*c^3*d^3*e^4 - A*c^3*d^2*e^5)*x^4 + 20*(7*B*c^3*d^4*e^3 - A*c^3*d^3*e^4)
*x^3 + 15*(7*B*c^3*d^5*e^2 - A*c^3*d^4*e^3)*x^2 + 6*(7*B*c^3*d^6*e - A*c^3*d^5*e^2)*x)*log(e*x + d))/(e^14*x^6
+ 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**7,x)

[Out]

Timed out

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Giac [A]  time = 1.15437, size = 575, normalized size = 1.8 \begin{align*} B c^{3} x e^{\left (-7\right )} -{\left (7 \, B c^{3} d - A c^{3} e\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (669 \, B c^{3} d^{7} - 147 \, A c^{3} d^{6} e + 30 \, 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} + 3 \, A a^{2} c d^{2} e^{5} + 180 \,{\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} + 2 \, B a^{3} d e^{6} + 30 \,{\left (175 \, B c^{3} d^{3} e^{4} - 45 \, 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 \, A a^{3} e^{7} + 20 \,{\left (455 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, 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} + 15 \,{\left (539 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, 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} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \,{\left (609 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, 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} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x\right )} e^{\left (-8\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^7,x, algorithm="giac")

[Out]

B*c^3*x*e^(-7) - (7*B*c^3*d - A*c^3*e)*e^(-8)*log(abs(x*e + d)) - 1/60*(669*B*c^3*d^7 - 147*A*c^3*d^6*e + 30*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 + 3*A*a^2*c*d^2*e^5 + 180*(7*B*c^3*d^2*e^5 - 2*A*c^3*d*
e^6 + B*a*c^2*e^7)*x^5 + 2*B*a^3*d*e^6 + 30*(175*B*c^3*d^3*e^4 - 45*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*A*a^3*e^7 + 20*(455*B*c^3*d^4*e^3 - 110*A*c^3*d^3*e^4 + 30*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 + 15*(539*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30*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 + 3*A*a^2*c*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 137*A*c^3*d^5*e^2 + 30*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 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x)*e^(-8)/(x*e + d)^6