3.1322 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^4} \, dx\)

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

[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

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Rubi [A]  time = 0.395793, antiderivative size = 310, 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} \[ -\frac{c \log (d+e x) \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac{c^2 x^2 \left (3 a B e^2-4 A c d e+10 B c d^2\right )}{2 e^6}-\frac{c^2 x \left (-3 a A e^3+12 a B d e^2-10 A c d^2 e+20 B c d^3\right )}{e^7}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^3}-\frac{c^3 x^3 (4 B d-A e)}{3 e^5}+\frac{B c^3 x^4}{4 e^4} \]

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*}

Mathematica [A]  time = 0.155664, size = 294, normalized size = 0.95 \[ \frac{12 c \log (d+e x) \left (B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )-4 A c d e \left (3 a e^2+5 c d^2\right )\right )+6 c^2 e^2 x^2 \left (3 a B e^2-4 A c d e+10 B c d^2\right )+12 c^2 e x \left (A e \left (3 a e^2+10 c d^2\right )-4 B \left (3 a d e^2+5 c d^3\right )\right )+\frac{36 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{d+e x}-\frac{6 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{(d+e x)^2}+\frac{4 \left (a e^2+c d^2\right )^3 (B d-A e)}{(d+e x)^3}+4 c^3 e^3 x^3 (A e-4 B d)+3 B c^3 e^4 x^4}{12 e^8} \]

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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Maple [B]  time = 0.013, size = 611, normalized size = 2. \begin{align*}{\frac{B{c}^{3}{x}^{4}}{4\,{e}^{4}}}-12\,{\frac{{c}^{2}\ln \left ( ex+d \right ) Ada}{{e}^{5}}}+30\,{\frac{{c}^{2}\ln \left ( ex+d \right ) Ba{d}^{2}}{{e}^{6}}}-18\,{\frac{A{d}^{2}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+9\,{\frac{B{a}^{2}cd}{{e}^{4} \left ( ex+d \right ) }}+30\,{\frac{Ba{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}+3\,{\frac{Ad{a}^{2}c}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{A{d}^{3}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{9\,B{a}^{2}c{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{15\,Ba{c}^{2}{d}^{4}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{A{d}^{2}{a}^{2}c}{{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{A{d}^{4}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{B{d}^{3}{a}^{2}c}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{Ba{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}-12\,{\frac{Ba{c}^{2}dx}{{e}^{5}}}-{\frac{B{a}^{3}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{A{a}^{3}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{A{x}^{3}{c}^{3}}{3\,{e}^{4}}}+{\frac{3\,B{x}^{2}a{c}^{2}}{2\,{e}^{4}}}+3\,{\frac{aA{c}^{2}x}{{e}^{4}}}-3\,{\frac{A{a}^{2}c}{{e}^{3} \left ( ex+d \right ) }}+{\frac{Bd{a}^{3}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+3\,{\frac{c\ln \left ( ex+d \right ) B{a}^{2}}{{e}^{4}}}-20\,{\frac{\ln \left ( ex+d \right ) A{c}^{3}{d}^{3}}{{e}^{7}}}+35\,{\frac{\ln \left ( ex+d \right ) B{c}^{3}{d}^{4}}{{e}^{8}}}-{\frac{7\,B{c}^{3}{d}^{6}}{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}-{\frac{4\,B{c}^{3}{x}^{3}d}{3\,{e}^{5}}}-{\frac{{d}^{6}A{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{B{c}^{3}{d}^{7}}{3\,{e}^{8} \left ( ex+d \right ) ^{3}}}-15\,{\frac{A{d}^{4}{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+21\,{\frac{B{c}^{3}{d}^{5}}{{e}^{8} \left ( ex+d \right ) }}-2\,{\frac{A{x}^{2}{c}^{3}d}{{e}^{5}}}+5\,{\frac{B{c}^{3}{x}^{2}{d}^{2}}{{e}^{6}}}+10\,{\frac{Ax{c}^{3}{d}^{2}}{{e}^{6}}}-20\,{\frac{B{c}^{3}{d}^{3}x}{{e}^{7}}}+3\,{\frac{A{d}^{5}{c}^{3}}{{e}^{7} \left ( ex+d \right ) ^{2}}} \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)^4,x)

[Out]

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

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Maxima [A]  time = 1.07959, size = 645, normalized size = 2.08 \begin{align*} \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{align*}

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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Fricas [B]  time = 2.02636, size = 1577, normalized size = 5.09 \begin{align*} \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{align*}

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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Sympy [A]  time = 92.7141, size = 524, normalized size = 1.69 \begin{align*} \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}} + \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}} - \frac{x^{3} \left (- A c^{3} e + 4 B c^{3} d\right )}{3 e^{5}} + \frac{x^{2} \left (- 4 A c^{3} d e + 3 B a c^{2} e^{2} + 10 B c^{3} d^{2}\right )}{2 e^{6}} - \frac{x \left (- 3 A a c^{2} e^{3} - 10 A c^{3} d^{2} e + 12 B a c^{2} d e^{2} + 20 B c^{3} d^{3}\right )}{e^{7}} \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)**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 + (-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) - x**3*(-A*c**3*e + 4*B*c**3*d)/(3*e**5) + x**2*(-4*A*c**3*d*e + 3*B*a*c**2*e**2 + 10*B*c**3*d**2)/(2
*e**6) - x*(-3*A*a*c**2*e**3 - 10*A*c**3*d**2*e + 12*B*a*c**2*d*e**2 + 20*B*c**3*d**3)/e**7

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Giac [A]  time = 1.21044, size = 587, normalized size = 1.89 \begin{align*}{\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{align*}

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