3.12.48 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=310 \[ -\frac {c \log (d+e x) \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac {c^2 x^2 \left (3 a B e^2-4 A c d e+10 B c d^2\right )}{2 e^6}-\frac {c^2 x \left (-3 a A e^3+12 a B d e^2-10 A c d^2 e+20 B c d^3\right )}{e^7}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^3}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)}-\frac {c^3 x^3 (4 B d-A e)}{3 e^5}+\frac {B c^3 x^4}{4 e^4} \]

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Rubi [A]  time = 0.40, antiderivative size = 310, 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 \log (d+e x) \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 )}{e^8}+\frac {c^2 x^2 \left (3 a B e^2-4 A c d e+10 B c d^2\right )}{2 e^6}-\frac {c^2 x \left (-3 a A e^3+12 a B d e^2-10 A c d^2 e+20 B c d^3\right )}{e^7}+\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 )}{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 )}{2 e^8 (d+e x)^2}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^3}-\frac {c^3 x^3 (4 B d-A e)}{3 e^5}+\frac {B c^3 x^4}{4 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^4,x]

[Out]

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

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Mathematica [A]  time = 0.13, size = 294, normalized size = 0.95 \begin {gather*} \frac {12 c \log (d+e x) \left (B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )-4 A c d e \left (3 a e^2+5 c d^2\right )\right )+6 c^2 e^2 x^2 \left (3 a B e^2-4 A c d e+10 B c d^2\right )+12 c^2 e x \left (A e \left (3 a e^2+10 c d^2\right )-4 B \left (3 a d e^2+5 c d^3\right )\right )-\frac {6 \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 )}{(d+e x)^2}+\frac {4 \left (a e^2+c d^2\right )^3 (B d-A e)}{(d+e x)^3}+\frac {36 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 )}{d+e x}+4 c^3 e^3 x^3 (A e-4 B d)+3 B c^3 e^4 x^4}{12 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^4,x]

[Out]

(12*c^2*e*(A*e*(10*c*d^2 + 3*a*e^2) - 4*B*(5*c*d^3 + 3*a*d*e^2))*x + 6*c^2*e^2*(10*B*c*d^2 - 4*A*c*d*e + 3*a*B
*e^2)*x^2 + 4*c^3*e^3*(-4*B*d + A*e)*x^3 + 3*B*c^3*e^4*x^4 + (4*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(d + e*x)^3 - (
6*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(d + e*x)^2 + (36*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) + 12*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))*Log[d + e*x])/(12*e^8)

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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^4, x]

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fricas [B]  time = 0.41, size = 732, normalized size = 2.36 \begin {gather*} \frac {3 \, B c^{3} e^{7} x^{7} + 214 \, B c^{3} d^{7} - 148 \, A c^{3} d^{6} e + 282 \, B a c^{2} d^{5} e^{2} - 156 \, A a c^{2} d^{4} e^{3} + 66 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 4 \, A a^{3} e^{7} - {\left (7 \, B c^{3} d e^{6} - 4 \, A c^{3} e^{7}\right )} x^{6} + 3 \, {\left (7 \, B c^{3} d^{2} e^{5} - 4 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} - 3 \, {\left (35 \, B c^{3} d^{3} e^{4} - 20 \, A c^{3} d^{2} e^{5} + 30 \, B a c^{2} d e^{6} - 12 \, A a c^{2} e^{7}\right )} x^{4} - 2 \, {\left (278 \, B c^{3} d^{4} e^{3} - 146 \, A c^{3} d^{3} e^{4} + 189 \, B a c^{2} d^{2} e^{5} - 54 \, A a c^{2} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, B c^{3} d^{5} e^{2} - 26 \, A c^{3} d^{4} e^{3} + 9 \, B a c^{2} d^{3} e^{4} + 18 \, A a c^{2} d^{2} e^{5} - 18 \, B a^{2} c d e^{6} + 6 \, A a^{2} c e^{7}\right )} x^{2} + 6 \, {\left (37 \, B c^{3} d^{6} e - 34 \, A c^{3} d^{5} e^{2} + 81 \, B a c^{2} d^{4} e^{3} - 54 \, A a c^{2} d^{3} e^{4} + 27 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x + 12 \, {\left (35 \, B c^{3} d^{7} - 20 \, A c^{3} d^{6} e + 30 \, 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} + {\left (35 \, B c^{3} d^{4} e^{3} - 20 \, A c^{3} d^{3} e^{4} + 30 \, 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} + 3 \, {\left (35 \, B c^{3} d^{5} e^{2} - 20 \, A c^{3} d^{4} e^{3} + 30 \, 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}\right )} x^{2} + 3 \, {\left (35 \, B c^{3} d^{6} e - 20 \, A c^{3} d^{5} e^{2} + 30 \, 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}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/12*(3*B*c^3*e^7*x^7 + 214*B*c^3*d^7 - 148*A*c^3*d^6*e + 282*B*a*c^2*d^5*e^2 - 156*A*a*c^2*d^4*e^3 + 66*B*a^2
*c*d^3*e^4 - 12*A*a^2*c*d^2*e^5 - 2*B*a^3*d*e^6 - 4*A*a^3*e^7 - (7*B*c^3*d*e^6 - 4*A*c^3*e^7)*x^6 + 3*(7*B*c^3
*d^2*e^5 - 4*A*c^3*d*e^6 + 6*B*a*c^2*e^7)*x^5 - 3*(35*B*c^3*d^3*e^4 - 20*A*c^3*d^2*e^5 + 30*B*a*c^2*d*e^6 - 12
*A*a*c^2*e^7)*x^4 - 2*(278*B*c^3*d^4*e^3 - 146*A*c^3*d^3*e^4 + 189*B*a*c^2*d^2*e^5 - 54*A*a*c^2*d*e^6)*x^3 - 6
*(68*B*c^3*d^5*e^2 - 26*A*c^3*d^4*e^3 + 9*B*a*c^2*d^3*e^4 + 18*A*a*c^2*d^2*e^5 - 18*B*a^2*c*d*e^6 + 6*A*a^2*c*
e^7)*x^2 + 6*(37*B*c^3*d^6*e - 34*A*c^3*d^5*e^2 + 81*B*a*c^2*d^4*e^3 - 54*A*a*c^2*d^3*e^4 + 27*B*a^2*c*d^2*e^5
- 6*A*a^2*c*d*e^6 - B*a^3*e^7)*x + 12*(35*B*c^3*d^7 - 20*A*c^3*d^6*e + 30*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 + (35*B*c^3*d^4*e^3 - 20*A*c^3*d^3*e^4 + 30*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 + 3*(35*B*c^3*d^5*e^2 - 20*A*c^3*d^4*e^3 + 30*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)*x^2 + 3*(35*B*c^3*d^6*e - 20*A*c^3*d^5*e^2 + 30*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)*
x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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giac [A]  time = 0.16, size = 435, normalized size = 1.40 \begin {gather*} {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B c^{3} x^{4} e^{12} - 16 \, B c^{3} d x^{3} e^{11} + 60 \, B c^{3} d^{2} x^{2} e^{10} - 240 \, B c^{3} d^{3} x e^{9} + 4 \, A c^{3} x^{3} e^{12} - 24 \, A c^{3} d x^{2} e^{11} + 120 \, A c^{3} d^{2} x e^{10} + 18 \, B a c^{2} x^{2} e^{12} - 144 \, B a c^{2} d x e^{11} + 36 \, A a c^{2} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, B c^{3} d^{7} - 74 \, A c^{3} d^{6} e + 141 \, B a c^{2} d^{5} e^{2} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a^{2} c d^{3} e^{4} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\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} + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, A c^{3} d^{5} e^{2} + 105 \, B a c^{2} d^{4} e^{3} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, 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 )}}{6 \, {\left (x e + d\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*e^(-8)*log(abs(x*e + d
)) + 1/12*(3*B*c^3*x^4*e^12 - 16*B*c^3*d*x^3*e^11 + 60*B*c^3*d^2*x^2*e^10 - 240*B*c^3*d^3*x*e^9 + 4*A*c^3*x^3*
e^12 - 24*A*c^3*d*x^2*e^11 + 120*A*c^3*d^2*x*e^10 + 18*B*a*c^2*x^2*e^12 - 144*B*a*c^2*d*x*e^11 + 36*A*a*c^2*x*
e^12)*e^(-16) + 1/6*(107*B*c^3*d^7 - 74*A*c^3*d^6*e + 141*B*a*c^2*d^5*e^2 - 78*A*a*c^2*d^4*e^3 + 33*B*a^2*c*d^
3*e^4 - 6*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 2*A*a^3*e^7 + 18*(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 + 3*(77*B*c^3*d^6*e - 54*A*c^3*d^5*e^2 + 105*B*a
*c^2*d^4*e^3 - 60*A*a*c^2*d^3*e^4 + 27*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)^3

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maple [B]  time = 0.07, size = 611, normalized size = 1.97 \begin {gather*} \frac {B \,c^{3} x^{4}}{4 e^{4}}-\frac {A \,a^{3}}{3 \left (e x +d \right )^{3} e}-\frac {A \,a^{2} c \,d^{2}}{\left (e x +d \right )^{3} e^{3}}-\frac {A a \,c^{2} d^{4}}{\left (e x +d \right )^{3} e^{5}}-\frac {A \,c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {A \,c^{3} x^{3}}{3 e^{4}}+\frac {B \,a^{3} d}{3 \left (e x +d \right )^{3} e^{2}}+\frac {B \,a^{2} c \,d^{3}}{\left (e x +d \right )^{3} e^{4}}+\frac {B a \,c^{2} d^{5}}{\left (e x +d \right )^{3} e^{6}}+\frac {B \,c^{3} d^{7}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {4 B \,c^{3} d \,x^{3}}{3 e^{5}}+\frac {3 A \,a^{2} c d}{\left (e x +d \right )^{2} e^{3}}+\frac {6 A a \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}+\frac {3 A \,c^{3} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {2 A \,c^{3} d \,x^{2}}{e^{5}}-\frac {B \,a^{3}}{2 \left (e x +d \right )^{2} e^{2}}-\frac {9 B \,a^{2} c \,d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {15 B a \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {3 B a \,c^{2} x^{2}}{2 e^{4}}-\frac {7 B \,c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {5 B \,c^{3} d^{2} x^{2}}{e^{6}}-\frac {3 A \,a^{2} c}{\left (e x +d \right ) e^{3}}-\frac {18 A a \,c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {12 A a \,c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {3 A a \,c^{2} x}{e^{4}}-\frac {15 A \,c^{3} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {20 A \,c^{3} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {10 A \,c^{3} d^{2} x}{e^{6}}+\frac {9 B \,a^{2} c d}{\left (e x +d \right ) e^{4}}+\frac {3 B \,a^{2} c \ln \left (e x +d \right )}{e^{4}}+\frac {30 B a \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {30 B a \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {12 B a \,c^{2} d x}{e^{5}}+\frac {21 B \,c^{3} d^{5}}{\left (e x +d \right ) e^{8}}+\frac {35 B \,c^{3} d^{4} \ln \left (e x +d \right )}{e^{8}}-\frac {20 B \,c^{3} d^{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)^4,x)

[Out]

-12*c^2/e^5*a*B*d*x+3/e^3/(e*x+d)^2*A*a^2*c*d+6/e^5/(e*x+d)^2*A*a*c^2*d^3-9/2/e^4/(e*x+d)^2*B*a^2*c*d^2-15/2/e
^6/(e*x+d)^2*B*a*c^2*d^4-1/e^3/(e*x+d)^3*A*d^2*a^2*c+3/e^7/(e*x+d)^2*A*c^3*d^5-7/2/e^8/(e*x+d)^2*B*c^3*d^6-1/2
/e^2/(e*x+d)^2*B*a^3-1/3/e/(e*x+d)^3*A*a^3+1/3*c^3/e^4*A*x^3-18*c^2/e^5/(e*x+d)*A*d^2*a+9*c/e^4/(e*x+d)*B*a^2*
d+30*c^2/e^6/(e*x+d)*B*d^3*a-12*c^2/e^5*ln(e*x+d)*A*d*a+30*c^2/e^6*ln(e*x+d)*B*d^2*a+1/e^4/(e*x+d)^3*B*d^3*a^2
*c+1/e^6/(e*x+d)^3*B*a*c^2*d^5-1/3/e^7/(e*x+d)^3*A*c^3*d^6-4/3*c^3/e^5*B*x^3*d-2*c^3/e^5*A*x^2*d-20*c^3/e^7*ln
(e*x+d)*A*d^3+3*c/e^4*ln(e*x+d)*B*a^2+35*c^3/e^8*ln(e*x+d)*B*d^4-1/e^5/(e*x+d)^3*A*a*c^2*d^4+3/2*c^2/e^4*B*x^2
*a+5*c^3/e^6*B*x^2*d^2+3*c^2/e^4*a*A*x+10*c^3/e^6*A*d^2*x-20*c^3/e^7*B*d^3*x-3*c/e^3/(e*x+d)*A*a^2-15*c^3/e^7/
(e*x+d)*A*d^4+21*c^3/e^8/(e*x+d)*B*d^5+1/3/e^8/(e*x+d)^3*B*c^3*d^7+1/3/e^2/(e*x+d)^3*B*d*a^3+1/4*B*c^3/e^4*x^4

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maxima [A]  time = 0.66, size = 478, normalized size = 1.54 \begin {gather*} \frac {107 \, B c^{3} d^{7} - 74 \, A c^{3} d^{6} e + 141 \, B a c^{2} d^{5} e^{2} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a^{2} c d^{3} e^{4} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\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} + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, A c^{3} d^{5} e^{2} + 105 \, B a c^{2} d^{4} e^{3} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, B c^{3} e^{3} x^{4} - 4 \, {\left (4 \, B c^{3} d e^{2} - A c^{3} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e - 4 \, A c^{3} d e^{2} + 3 \, B a c^{2} e^{3}\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - 10 \, A c^{3} d^{2} e + 12 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} x}{12 \, e^{7}} + \frac {{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\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)^4,x, algorithm="maxima")

[Out]

1/6*(107*B*c^3*d^7 - 74*A*c^3*d^6*e + 141*B*a*c^2*d^5*e^2 - 78*A*a*c^2*d^4*e^3 + 33*B*a^2*c*d^3*e^4 - 6*A*a^2*
c*d^2*e^5 - B*a^3*d*e^6 - 2*A*a^3*e^7 + 18*(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 + 3*(77*B*c^3*d^6*e - 54*A*c^3*d^5*e^2 + 105*B*a*c^2*d^4*e^3 - 6
0*A*a*c^2*d^3*e^4 + 27*B*a^2*c*d^2*e^5 - 6*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*
x + d^3*e^8) + 1/12*(3*B*c^3*e^3*x^4 - 4*(4*B*c^3*d*e^2 - A*c^3*e^3)*x^3 + 6*(10*B*c^3*d^2*e - 4*A*c^3*d*e^2 +
3*B*a*c^2*e^3)*x^2 - 12*(20*B*c^3*d^3 - 10*A*c^3*d^2*e + 12*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*x)/e^7 + (35*B*c^3
*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*log(e*x + d)/e^8

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mupad [B]  time = 1.78, size = 548, normalized size = 1.77 \begin {gather*} x^3\,\left (\frac {A\,c^3}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )-\frac {\frac {B\,a^3\,d\,e^6+2\,A\,a^3\,e^7-33\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-141\,B\,a\,c^2\,d^5\,e^2+78\,A\,a\,c^2\,d^4\,e^3-107\,B\,c^3\,d^7+74\,A\,c^3\,d^6\,e}{6\,e}+x^2\,\left (-9\,B\,a^2\,c\,d\,e^5+3\,A\,a^2\,c\,e^6-30\,B\,a\,c^2\,d^3\,e^3+18\,A\,a\,c^2\,d^2\,e^4-21\,B\,c^3\,d^5\,e+15\,A\,c^3\,d^4\,e^2\right )+x\,\left (\frac {B\,a^3\,e^6}{2}-\frac {27\,B\,a^2\,c\,d^2\,e^4}{2}+3\,A\,a^2\,c\,d\,e^5-\frac {105\,B\,a\,c^2\,d^4\,e^2}{2}+30\,A\,a\,c^2\,d^3\,e^3-\frac {77\,B\,c^3\,d^6}{2}+27\,A\,c^3\,d^5\,e\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x\,\left (\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,a\,c^2}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {A\,c^3}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^4}-\frac {4\,B\,c^3\,d^3}{e^7}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,a\,c^2}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,c\,e^4+30\,B\,a\,c^2\,d^2\,e^2-12\,A\,a\,c^2\,d\,e^3+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \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)^4,x)

[Out]

x^3*((A*c^3)/(3*e^4) - (4*B*c^3*d)/(3*e^5)) - ((2*A*a^3*e^7 - 107*B*c^3*d^7 + B*a^3*d*e^6 + 74*A*c^3*d^6*e + 7
8*A*a*c^2*d^4*e^3 + 6*A*a^2*c*d^2*e^5 - 141*B*a*c^2*d^5*e^2 - 33*B*a^2*c*d^3*e^4)/(6*e) + x^2*(3*A*a^2*c*e^6 -
21*B*c^3*d^5*e + 15*A*c^3*d^4*e^2 + 18*A*a*c^2*d^2*e^4 - 30*B*a*c^2*d^3*e^3 - 9*B*a^2*c*d*e^5) + x*((B*a^3*e^
6)/2 - (77*B*c^3*d^6)/2 + 27*A*c^3*d^5*e + 30*A*a*c^2*d^3*e^3 - (105*B*a*c^2*d^4*e^2)/2 - (27*B*a^2*c*d^2*e^4)
/2 + 3*A*a^2*c*d*e^5))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) + x*((4*d*((4*d*((A*c^3)/e^4 - (4*B*c^
3*d)/e^5))/e - (3*B*a*c^2)/e^4 + (6*B*c^3*d^2)/e^6))/e - (6*d^2*((A*c^3)/e^4 - (4*B*c^3*d)/e^5))/e^2 + (3*A*a*
c^2)/e^4 - (4*B*c^3*d^3)/e^7) - x^2*((2*d*((A*c^3)/e^4 - (4*B*c^3*d)/e^5))/e - (3*B*a*c^2)/(2*e^4) + (3*B*c^3*
d^2)/e^6) + (log(d + e*x)*(35*B*c^3*d^4 + 3*B*a^2*c*e^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e
^3))/e^8 + (B*c^3*x^4)/(4*e^4)

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sympy [A]  time = 18.98, size = 530, normalized size = 1.71 \begin {gather*} \frac {B c^{3} x^{4}}{4 e^{4}} + \frac {c \left (- 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}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{3} \left (\frac {A c^{3}}{3 e^{4}} - \frac {4 B c^{3} d}{3 e^{5}}\right ) + x^{2} \left (- \frac {2 A c^{3} d}{e^{5}} + \frac {3 B a c^{2}}{2 e^{4}} + \frac {5 B c^{3} d^{2}}{e^{6}}\right ) + x \left (\frac {3 A a c^{2}}{e^{4}} + \frac {10 A c^{3} d^{2}}{e^{6}} - \frac {12 B a c^{2} d}{e^{5}} - \frac {20 B c^{3} d^{3}}{e^{7}}\right ) + \frac {- 2 A a^{3} e^{7} - 6 A a^{2} c d^{2} e^{5} - 78 A a c^{2} d^{4} e^{3} - 74 A c^{3} d^{6} e - B a^{3} d e^{6} + 33 B a^{2} c d^{3} e^{4} + 141 B a c^{2} d^{5} e^{2} + 107 B c^{3} d^{7} + x^{2} \left (- 18 A a^{2} c e^{7} - 108 A a c^{2} d^{2} e^{5} - 90 A c^{3} d^{4} e^{3} + 54 B a^{2} c d e^{6} + 180 B a c^{2} d^{3} e^{4} + 126 B c^{3} d^{5} e^{2}\right ) + x \left (- 18 A a^{2} c d e^{6} - 180 A a c^{2} d^{3} e^{4} - 162 A c^{3} d^{5} e^{2} - 3 B a^{3} e^{7} + 81 B a^{2} c d^{2} e^{5} + 315 B a c^{2} d^{4} e^{3} + 231 B c^{3} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} \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)**4,x)

[Out]

B*c**3*x**4/(4*e**4) + 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)*log(d + e*x)/e**8 + x**3*(A*c**3/(3*e**4) - 4*B*c**3*d/(3*e**5)) + x**2*(-2*A*c**3*d/e**5 + 3*B*a*c**2/
(2*e**4) + 5*B*c**3*d**2/e**6) + x*(3*A*a*c**2/e**4 + 10*A*c**3*d**2/e**6 - 12*B*a*c**2*d/e**5 - 20*B*c**3*d**
3/e**7) + (-2*A*a**3*e**7 - 6*A*a**2*c*d**2*e**5 - 78*A*a*c**2*d**4*e**3 - 74*A*c**3*d**6*e - B*a**3*d*e**6 +
33*B*a**2*c*d**3*e**4 + 141*B*a*c**2*d**5*e**2 + 107*B*c**3*d**7 + x**2*(-18*A*a**2*c*e**7 - 108*A*a*c**2*d**2
*e**5 - 90*A*c**3*d**4*e**3 + 54*B*a**2*c*d*e**6 + 180*B*a*c**2*d**3*e**4 + 126*B*c**3*d**5*e**2) + x*(-18*A*a
**2*c*d*e**6 - 180*A*a*c**2*d**3*e**4 - 162*A*c**3*d**5*e**2 - 3*B*a**3*e**7 + 81*B*a**2*c*d**2*e**5 + 315*B*a
*c**2*d**4*e**3 + 231*B*c**3*d**6*e))/(6*d**3*e**8 + 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3)

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