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

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

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Rubi [A]  time = 0.32, antiderivative size = 320, 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 )}{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} \end {gather*}

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*}

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Mathematica [A]  time = 0.21, size = 377, normalized size = 1.18 \begin {gather*} -\frac {A e \left (10 a^3 e^6+3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )+6 a c^2 e^2 \left (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\right )-c^3 d \left (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\right )\right )+B \left (2 a^3 e^6 (d+6 e x)+3 a^2 c e^4 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+30 a c^2 e^2 \left (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\right )+c^3 \left (669 d^7+3594 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\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(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
) + 30*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
+ 3594*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 - 6
0*e^7*x^7)) + 60*c^3*(7*B*d - A*e)*(d + e*x)^6*Log[d + e*x])/(e^8*(d + e*x)^6)

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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)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 695, normalized size = 2.17 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 426, normalized size = 1.33 \begin {gather*} 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 {gather*}

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

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maple [B]  time = 0.06, size = 656, normalized size = 2.05 \begin {gather*} -\frac {A \,a^{3}}{6 \left (e x +d \right )^{6} e}-\frac {A \,a^{2} c \,d^{2}}{2 \left (e x +d \right )^{6} e^{3}}-\frac {A a \,c^{2} d^{4}}{2 \left (e x +d \right )^{6} e^{5}}-\frac {A \,c^{3} d^{6}}{6 \left (e x +d \right )^{6} e^{7}}+\frac {B \,a^{3} d}{6 \left (e x +d \right )^{6} e^{2}}+\frac {B \,a^{2} c \,d^{3}}{2 \left (e x +d \right )^{6} e^{4}}+\frac {B a \,c^{2} d^{5}}{2 \left (e x +d \right )^{6} e^{6}}+\frac {B \,c^{3} d^{7}}{6 \left (e x +d \right )^{6} e^{8}}+\frac {6 A \,a^{2} c d}{5 \left (e x +d \right )^{5} e^{3}}+\frac {12 A a \,c^{2} d^{3}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {6 A \,c^{3} d^{5}}{5 \left (e x +d \right )^{5} e^{7}}-\frac {B \,a^{3}}{5 \left (e x +d \right )^{5} e^{2}}-\frac {9 B \,a^{2} c \,d^{2}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {3 B a \,c^{2} d^{4}}{\left (e x +d \right )^{5} e^{6}}-\frac {7 B \,c^{3} d^{6}}{5 \left (e x +d \right )^{5} e^{8}}-\frac {3 A \,a^{2} c}{4 \left (e x +d \right )^{4} e^{3}}-\frac {9 A a \,c^{2} d^{2}}{2 \left (e x +d \right )^{4} e^{5}}-\frac {15 A \,c^{3} d^{4}}{4 \left (e x +d \right )^{4} e^{7}}+\frac {9 B \,a^{2} c d}{4 \left (e x +d \right )^{4} e^{4}}+\frac {15 B a \,c^{2} d^{3}}{2 \left (e x +d \right )^{4} e^{6}}+\frac {21 B \,c^{3} d^{5}}{4 \left (e x +d \right )^{4} e^{8}}+\frac {4 A a \,c^{2} d}{\left (e x +d \right )^{3} e^{5}}+\frac {20 A \,c^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{7}}-\frac {B \,a^{2} c}{\left (e x +d \right )^{3} e^{4}}-\frac {10 B a \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{6}}-\frac {35 B \,c^{3} d^{4}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {3 A a \,c^{2}}{2 \left (e x +d \right )^{2} e^{5}}-\frac {15 A \,c^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {15 B a \,c^{2} d}{2 \left (e x +d \right )^{2} e^{6}}+\frac {35 B \,c^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {6 A \,c^{3} d}{\left (e x +d \right ) e^{7}}+\frac {A \,c^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {3 B a \,c^{2}}{\left (e x +d \right ) e^{6}}-\frac {21 B \,c^{3} d^{2}}{\left (e x +d \right ) e^{8}}-\frac {7 B \,c^{3} d \ln \left (e x +d \right )}{e^{8}}+\frac {B \,c^{3} 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)^7,x)

[Out]

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

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maxima [A]  time = 0.77, size = 511, normalized size = 1.60 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.86, size = 505, normalized size = 1.58 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c^3\,e-7\,B\,c^3\,d\right )}{e^8}-\frac {\frac {2\,B\,a^3\,d\,e^6+10\,A\,a^3\,e^7+3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5+30\,B\,a\,c^2\,d^5\,e^2+6\,A\,a\,c^2\,d^4\,e^3+669\,B\,c^3\,d^7-147\,A\,c^3\,d^6\,e}{60\,e}+x^2\,\left (\frac {3\,B\,a^2\,c\,d\,e^5}{4}+\frac {3\,A\,a^2\,c\,e^6}{4}+\frac {15\,B\,a\,c^2\,d^3\,e^3}{2}+\frac {3\,A\,a\,c^2\,d^2\,e^4}{2}+\frac {539\,B\,c^3\,d^5\,e}{4}-\frac {125\,A\,c^3\,d^4\,e^2}{4}\right )+x^3\,\left (B\,a^2\,c\,e^6+10\,B\,a\,c^2\,d^2\,e^4+2\,A\,a\,c^2\,d\,e^5+\frac {455\,B\,c^3\,d^4\,e^2}{3}-\frac {110\,A\,c^3\,d^3\,e^3}{3}\right )+x^5\,\left (21\,B\,c^3\,d^2\,e^4-6\,A\,c^3\,d\,e^5+3\,B\,a\,c^2\,e^6\right )+x\,\left (\frac {B\,a^3\,e^6}{5}+\frac {3\,B\,a^2\,c\,d^2\,e^4}{10}+\frac {3\,A\,a^2\,c\,d\,e^5}{10}+3\,B\,a\,c^2\,d^4\,e^2+\frac {3\,A\,a\,c^2\,d^3\,e^3}{5}+\frac {609\,B\,c^3\,d^6}{10}-\frac {137\,A\,c^3\,d^5\,e}{10}\right )+x^4\,\left (\frac {175\,B\,c^3\,d^3\,e^3}{2}-\frac {45\,A\,c^3\,d^2\,e^4}{2}+\frac {15\,B\,a\,c^2\,d\,e^5}{2}+\frac {3\,A\,a\,c^2\,e^6}{2}\right )}{d^6\,e^7+6\,d^5\,e^8\,x+15\,d^4\,e^9\,x^2+20\,d^3\,e^{10}\,x^3+15\,d^2\,e^{11}\,x^4+6\,d\,e^{12}\,x^5+e^{13}\,x^6}+\frac {B\,c^3\,x}{e^7} \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)^7,x)

[Out]

(log(d + e*x)*(A*c^3*e - 7*B*c^3*d))/e^8 - ((10*A*a^3*e^7 + 669*B*c^3*d^7 + 2*B*a^3*d*e^6 - 147*A*c^3*d^6*e +
6*A*a*c^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 + 30*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4)/(60*e) + x^2*((3*A*a^2*c*e^6)/
4 + (539*B*c^3*d^5*e)/4 - (125*A*c^3*d^4*e^2)/4 + (3*A*a*c^2*d^2*e^4)/2 + (15*B*a*c^2*d^3*e^3)/2 + (3*B*a^2*c*
d*e^5)/4) + x^3*(B*a^2*c*e^6 - (110*A*c^3*d^3*e^3)/3 + (455*B*c^3*d^4*e^2)/3 + 10*B*a*c^2*d^2*e^4 + 2*A*a*c^2*
d*e^5) + x^5*(3*B*a*c^2*e^6 - 6*A*c^3*d*e^5 + 21*B*c^3*d^2*e^4) + x*((B*a^3*e^6)/5 + (609*B*c^3*d^6)/10 - (137
*A*c^3*d^5*e)/10 + (3*A*a*c^2*d^3*e^3)/5 + 3*B*a*c^2*d^4*e^2 + (3*B*a^2*c*d^2*e^4)/10 + (3*A*a^2*c*d*e^5)/10)
+ x^4*((3*A*a*c^2*e^6)/2 - (45*A*c^3*d^2*e^4)/2 + (175*B*c^3*d^3*e^3)/2 + (15*B*a*c^2*d*e^5)/2))/(d^6*e^7 + e^
13*x^6 + 6*d^5*e^8*x + 6*d*e^12*x^5 + 15*d^4*e^9*x^2 + 20*d^3*e^10*x^3 + 15*d^2*e^11*x^4) + (B*c^3*x)/e^7

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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)**7,x)

[Out]

Timed out

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