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*(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

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Rubi [A]  time = 0.416529, antiderivative size = 300, 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} \[ -\frac{c x \left (A c d e \left (9 a e^2+10 c d^2\right )-3 B \left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}+\frac{c^2 x^3 \left (a B e^2-A c d e+2 B c d^2\right )}{e^5}-\frac{c^2 x^2 \left (-3 a A e^3+9 a B d e^2-6 A c d^2 e+10 B c d^3\right )}{2 e^6}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{2 e^8 (d+e x)^2}-\frac{3 c \left (a e^2+c d^2\right ) \log (d+e x) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac{c^3 x^4 (3 B d-A e)}{4 e^4}+\frac{B c^3 x^5}{5 e^3} \]

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*}

Mathematica [A]  time = 0.204847, size = 414, normalized size = 1.38 \[ \frac{5 A e \left (6 a^2 c d e^4 (3 d+4 e x)-2 a^3 e^6+6 a c^2 e^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )\right )+B \left (30 a^2 c e^4 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )-10 a^3 e^6 (d+2 e x)+10 a c^2 e^2 \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )+c^3 \left (500 d^5 e^2 x^2+140 d^4 e^3 x^3-35 d^3 e^4 x^4+14 d^2 e^5 x^5+160 d^6 e x-130 d^7-7 d e^6 x^6+4 e^7 x^7\right )\right )-60 c (d+e x)^2 \left (a e^2+c d^2\right ) \log (d+e x) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{20 e^8 (d+e x)^2} \]

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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Maple [B]  time = 0.012, size = 589, normalized size = 2. \begin{align*}{\frac{3\,Ba{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+18\,{\frac{Ba{c}^{2}{d}^{2}x}{{e}^{5}}}+6\,{\frac{Ad{a}^{2}c}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{A{d}^{3}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-9\,{\frac{B{a}^{2}c{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}-15\,{\frac{Ba{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{3\,A{d}^{2}{a}^{2}c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,A{d}^{4}a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{3\,B{a}^{2}c{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{3}{x}^{5}}{5\,{e}^{3}}}+18\,{\frac{{c}^{2}\ln \left ( ex+d \right ) A{d}^{2}a}{{e}^{5}}}-9\,{\frac{c\ln \left ( ex+d \right ) B{a}^{2}d}{{e}^{4}}}-30\,{\frac{{c}^{2}\ln \left ( ex+d \right ) aB{d}^{3}}{{e}^{6}}}-{\frac{9\,B{x}^{2}a{c}^{2}d}{2\,{e}^{4}}}-9\,{\frac{Ada{c}^{2}x}{{e}^{4}}}-{\frac{A{a}^{3}}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{B{a}^{3}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{A{c}^{3}{x}^{4}}{4\,{e}^{3}}}+3\,{\frac{{a}^{2}Bcx}{{e}^{3}}}+{\frac{Bd{a}^{3}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{aB{c}^{2}{x}^{3}}{{e}^{3}}}+3\,{\frac{c\ln \left ( ex+d \right ) A{a}^{2}}{{e}^{3}}}+{\frac{3\,aA{c}^{2}{x}^{2}}{2\,{e}^{3}}}-10\,{\frac{A{c}^{3}{d}^{3}x}{{e}^{6}}}+15\,{\frac{B{c}^{3}{d}^{4}x}{{e}^{7}}}+15\,{\frac{{d}^{4}\ln \left ( ex+d \right ) A{c}^{3}}{{e}^{7}}}-{\frac{A{x}^{3}{c}^{3}d}{{e}^{4}}}-7\,{\frac{B{c}^{3}{d}^{6}}{{e}^{8} \left ( ex+d \right ) }}-21\,{\frac{{d}^{5}\ln \left ( ex+d \right ) B{c}^{3}}{{e}^{8}}}-{\frac{{d}^{6}A{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{3}{d}^{7}}{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}+2\,{\frac{B{c}^{3}{x}^{3}{d}^{2}}{{e}^{5}}}+3\,{\frac{A{x}^{2}{c}^{3}{d}^{2}}{{e}^{5}}}-5\,{\frac{B{c}^{3}{x}^{2}{d}^{3}}{{e}^{6}}}-{\frac{3\,B{c}^{3}{x}^{4}d}{4\,{e}^{4}}}+6\,{\frac{A{d}^{5}{c}^{3}}{{e}^{7} \left ( ex+d \right ) }} \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)^3,x)

[Out]

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

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Maxima [A]  time = 1.13318, size = 626, normalized size = 2.09 \begin{align*} -\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}} \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)^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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Fricas [B]  time = 1.8526, size = 1461, normalized size = 4.87 \begin{align*} \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 )}} \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)^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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Sympy [A]  time = 14.0657, size = 479, normalized size = 1.6 \begin{align*} \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}} - \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}} - \frac{x^{4} \left (- A c^{3} e + 3 B c^{3} d\right )}{4 e^{4}} + \frac{x^{3} \left (- A c^{3} d e + B a c^{2} e^{2} + 2 B c^{3} d^{2}\right )}{e^{5}} - \frac{x^{2} \left (- 3 A a c^{2} e^{3} - 6 A c^{3} d^{2} e + 9 B a c^{2} d e^{2} + 10 B c^{3} d^{3}\right )}{2 e^{6}} + \frac{x \left (- 9 A a c^{2} d e^{3} - 10 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 18 B a c^{2} d^{2} e^{2} + 15 B c^{3} d^{4}\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)**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 - (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) - x**4*(-A*c**3*e + 3*B*c**3*d)/(4*e**4) + x**3*(-A*c**3*d*e + B*a*c**2*
e**2 + 2*B*c**3*d**2)/e**5 - x**2*(-3*A*a*c**2*e**3 - 6*A*c**3*d**2*e + 9*B*a*c**2*d*e**2 + 10*B*c**3*d**3)/(2
*e**6) + x*(-9*A*a*c**2*d*e**3 - 10*A*c**3*d**3*e + 3*B*a**2*c*e**4 + 18*B*a*c**2*d**2*e**2 + 15*B*c**3*d**4)/
e**7

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Giac [A]  time = 1.15812, size = 594, normalized size = 1.98 \begin{align*} -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}} \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)^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