3.1320 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=309 \[ -\frac{c x^2 \left (2 A c d e \left (3 a e^2+2 c d^2\right )-B \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{2 e^6}-\frac{c x \left (6 B d \left (a e^2+c d^2\right )^2-A e \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}-\frac{c^2 x^4 \left (2 A c d e-3 B \left (a e^2+c d^2\right )\right )}{4 e^4}+\frac{c^2 x^3 \left (3 A e \left (a e^2+c d^2\right )-B \left (6 a d e^2+4 c d^3\right )\right )}{3 e^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 (d+e x)}+\frac{\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}-\frac{c^3 x^5 (2 B d-A e)}{5 e^3}+\frac{B c^3 x^6}{6 e^2} \]

[Out]

-((c*(6*B*d*(c*d^2 + a*e^2)^2 - A*e*(5*c^2*d^4 + 9*a*c*d^2*e^2 + 3*a^2*e^4))*x)/e^7) - (c*(2*A*c*d*e*(2*c*d^2
+ 3*a*e^2) - B*(5*c^2*d^4 + 9*a*c*d^2*e^2 + 3*a^2*e^4))*x^2)/(2*e^6) + (c^2*(3*A*e*(c*d^2 + a*e^2) - B*(4*c*d^
3 + 6*a*d*e^2))*x^3)/(3*e^5) - (c^2*(2*A*c*d*e - 3*B*(c*d^2 + a*e^2))*x^4)/(4*e^4) - (c^3*(2*B*d - A*e)*x^5)/(
5*e^3) + (B*c^3*x^6)/(6*e^2) + ((B*d - A*e)*(c*d^2 + a*e^2)^3)/(e^8*(d + e*x)) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2
- 6*A*c*d*e + a*B*e^2)*Log[d + e*x])/e^8

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Rubi [A]  time = 0.43546, antiderivative size = 309, 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} \[ -\frac{c x^2 \left (2 A c d e \left (3 a e^2+2 c d^2\right )-B \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{2 e^6}-\frac{c x \left (6 B d \left (a e^2+c d^2\right )^2-A e \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}-\frac{c^2 x^4 \left (2 A c d e-3 B \left (a e^2+c d^2\right )\right )}{4 e^4}+\frac{c^2 x^3 \left (3 A e \left (a e^2+c d^2\right )-B \left (6 a d e^2+4 c d^3\right )\right )}{3 e^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 (d+e x)}+\frac{\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}-\frac{c^3 x^5 (2 B d-A e)}{5 e^3}+\frac{B c^3 x^6}{6 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*(6*B*d*(c*d^2 + a*e^2)^2 - A*e*(5*c^2*d^4 + 9*a*c*d^2*e^2 + 3*a^2*e^4))*x)/e^7) - (c*(2*A*c*d*e*(2*c*d^2
+ 3*a*e^2) - B*(5*c^2*d^4 + 9*a*c*d^2*e^2 + 3*a^2*e^4))*x^2)/(2*e^6) + (c^2*(3*A*e*(c*d^2 + a*e^2) - B*(4*c*d^
3 + 6*a*d*e^2))*x^3)/(3*e^5) - (c^2*(2*A*c*d*e - 3*B*(c*d^2 + a*e^2))*x^4)/(4*e^4) - (c^3*(2*B*d - A*e)*x^5)/(
5*e^3) + (B*c^3*x^6)/(6*e^2) + ((B*d - A*e)*(c*d^2 + a*e^2)^3)/(e^8*(d + e*x)) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2
- 6*A*c*d*e + a*B*e^2)*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)^2} \, dx &=\int \left (\frac{c \left (-6 B d \left (c d^2+a e^2\right )^2+A e \left (5 c^2 d^4+9 a c d^2 e^2+3 a^2 e^4\right )\right )}{e^7}-\frac{c \left (-5 B c^2 d^4+4 A c^2 d^3 e-9 a B c d^2 e^2+6 a A c d e^3-3 a^2 B e^4\right ) x}{e^6}+\frac{c^2 \left (-4 B c d^3+3 A c d^2 e-6 a B d e^2+3 a A e^3\right ) x^2}{e^5}+\frac{c^2 \left (-2 A c d e+3 B \left (c d^2+a e^2\right )\right ) x^3}{e^4}+\frac{c^3 (-2 B d+A e) x^4}{e^3}+\frac{B c^3 x^5}{e^2}+\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (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^7 (d+e x)}\right ) \, dx\\ &=-\frac{c \left (6 B d \left (c d^2+a e^2\right )^2-A e \left (5 c^2 d^4+9 a c d^2 e^2+3 a^2 e^4\right )\right ) x}{e^7}-\frac{c \left (2 A c d e \left (2 c d^2+3 a e^2\right )-B \left (5 c^2 d^4+9 a c d^2 e^2+3 a^2 e^4\right )\right ) x^2}{2 e^6}+\frac{c^2 \left (3 A e \left (c d^2+a e^2\right )-B \left (4 c d^3+6 a d e^2\right )\right ) x^3}{3 e^5}-\frac{c^2 \left (2 A c d e-3 B \left (c d^2+a e^2\right )\right ) x^4}{4 e^4}-\frac{c^3 (2 B d-A e) x^5}{5 e^3}+\frac{B c^3 x^6}{6 e^2}+\frac{(B d-A e) \left (c d^2+a e^2\right )^3}{e^8 (d+e x)}+\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 ) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.188598, size = 405, normalized size = 1.31 \[ \frac{6 A e \left (30 a^2 c e^4 \left (-d^2+d e x+e^2 x^2\right )-10 a^3 e^6+10 a c^2 e^2 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+c^3 \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )\right )+B \left (90 a^2 c e^4 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+60 a^3 d e^6+15 a c^2 e^2 \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )+c^3 \left (-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-360 d^6 e x+60 d^7-14 d e^6 x^6+10 e^7 x^7\right )\right )+60 (d+e x) \left (a e^2+c d^2\right )^2 \log (d+e x) \left (a B e^2-6 A c d e+7 B c d^2\right )}{60 e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*A*e*(-10*a^3*e^6 + 30*a^2*c*e^4*(-d^2 + d*e*x + e^2*x^2) + 10*a*c^2*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2
- 2*d*e^3*x^3 + e^4*x^4) + c^3*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*
e^5*x^5 + 2*e^6*x^6)) + B*(60*a^3*d*e^6 + 90*a^2*c*e^4*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 15*a*c^2*
e^2*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + c^3*(60*d^7 - 360*d^6*
e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) + 60*(c
*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)*Log[d + e*x])/(60*e^8*(d + e*x))

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Maple [A]  time = 0.012, size = 558, normalized size = 1.8 \begin{align*}{\frac{B{c}^{3}{x}^{6}}{6\,{e}^{2}}}-12\,{\frac{\ln \left ( ex+d \right ) Aa{c}^{2}{d}^{3}}{{e}^{5}}}+9\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}c{d}^{2}}{{e}^{4}}}+15\,{\frac{\ln \left ( ex+d \right ) Ba{c}^{2}{d}^{4}}{{e}^{6}}}-2\,{\frac{aB{c}^{2}{x}^{3}d}{{e}^{3}}}-3\,{\frac{aA{c}^{2}{x}^{2}d}{{e}^{3}}}+{\frac{9\,B{x}^{2}a{c}^{2}{d}^{2}}{2\,{e}^{4}}}+9\,{\frac{A{d}^{2}a{c}^{2}x}{{e}^{4}}}-6\,{\frac{B{a}^{2}cdx}{{e}^{3}}}-12\,{\frac{Ba{c}^{2}{d}^{3}x}{{e}^{5}}}-3\,{\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{a}^{2}c{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+3\,{\frac{Ba{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-6\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}cd}{{e}^{3}}}-{\frac{A{a}^{3}}{e \left ( ex+d \right ) }}+{\frac{\ln \left ( ex+d \right ) B{a}^{3}}{{e}^{2}}}+{\frac{A{c}^{3}{x}^{5}}{5\,{e}^{2}}}+3\,{\frac{{a}^{2}Acx}{{e}^{2}}}+{\frac{aA{c}^{2}{x}^{3}}{{e}^{2}}}+{\frac{3\,aB{c}^{2}{x}^{4}}{4\,{e}^{2}}}+{\frac{Bd{a}^{3}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{3\,{a}^{2}Bc{x}^{2}}{2\,{e}^{2}}}-6\,{\frac{B{c}^{3}{d}^{5}x}{{e}^{7}}}+{\frac{A{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}+{\frac{5\,B{c}^{3}{x}^{2}{d}^{4}}{2\,{e}^{6}}}-2\,{\frac{A{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}+{\frac{3\,B{c}^{3}{x}^{4}{d}^{2}}{4\,{e}^{4}}}-{\frac{4\,B{c}^{3}{x}^{3}{d}^{3}}{3\,{e}^{5}}}-{\frac{A{c}^{3}{x}^{4}d}{2\,{e}^{3}}}-{\frac{2\,B{c}^{3}{x}^{5}d}{5\,{e}^{3}}}-6\,{\frac{{d}^{5}\ln \left ( ex+d \right ) A{c}^{3}}{{e}^{7}}}+7\,{\frac{{d}^{6}\ln \left ( ex+d \right ) B{c}^{3}}{{e}^{8}}}-{\frac{{d}^{6}A{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+5\,{\frac{A{d}^{4}{c}^{3}x}{{e}^{6}}}+{\frac{B{c}^{3}{d}^{7}}{{e}^{8} \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)^2,x)

[Out]

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

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Maxima [A]  time = 1.10359, size = 616, normalized size = 1.99 \begin{align*} \frac{B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, 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} + B a^{3} d e^{6} - A a^{3} e^{7}}{e^{9} x + d e^{8}} + \frac{10 \, B c^{3} e^{5} x^{6} - 12 \,{\left (2 \, B c^{3} d e^{4} - A c^{3} e^{5}\right )} x^{5} + 15 \,{\left (3 \, B c^{3} d^{2} e^{3} - 2 \, A c^{3} d e^{4} + 3 \, B a c^{2} e^{5}\right )} x^{4} - 20 \,{\left (4 \, B c^{3} d^{3} e^{2} - 3 \, A c^{3} d^{2} e^{3} + 6 \, B a c^{2} d e^{4} - 3 \, A a c^{2} e^{5}\right )} x^{3} + 30 \,{\left (5 \, B c^{3} d^{4} e - 4 \, A c^{3} d^{3} e^{2} + 9 \, B a c^{2} d^{2} e^{3} - 6 \, A a c^{2} d e^{4} + 3 \, B a^{2} c e^{5}\right )} x^{2} - 60 \,{\left (6 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 12 \, B a c^{2} d^{3} e^{2} - 9 \, A a c^{2} d^{2} e^{3} + 6 \, B a^{2} c d e^{4} - 3 \, A a^{2} c e^{5}\right )} x}{60 \, e^{7}} + \frac{{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\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)^2,x, algorithm="maxima")

[Out]

(B*c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*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 + B*a
^3*d*e^6 - A*a^3*e^7)/(e^9*x + d*e^8) + 1/60*(10*B*c^3*e^5*x^6 - 12*(2*B*c^3*d*e^4 - A*c^3*e^5)*x^5 + 15*(3*B*
c^3*d^2*e^3 - 2*A*c^3*d*e^4 + 3*B*a*c^2*e^5)*x^4 - 20*(4*B*c^3*d^3*e^2 - 3*A*c^3*d^2*e^3 + 6*B*a*c^2*d*e^4 - 3
*A*a*c^2*e^5)*x^3 + 30*(5*B*c^3*d^4*e - 4*A*c^3*d^3*e^2 + 9*B*a*c^2*d^2*e^3 - 6*A*a*c^2*d*e^4 + 3*B*a^2*c*e^5)
*x^2 - 60*(6*B*c^3*d^5 - 5*A*c^3*d^4*e + 12*B*a*c^2*d^3*e^2 - 9*A*a*c^2*d^2*e^3 + 6*B*a^2*c*d*e^4 - 3*A*a^2*c*
e^5)*x)/e^7 + (7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A
*a^2*c*d*e^5 + B*a^3*e^6)*log(e*x + d)/e^8

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Fricas [B]  time = 1.83837, size = 1331, normalized size = 4.31 \begin{align*} \frac{10 \, B c^{3} e^{7} x^{7} + 60 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e + 180 \, B a c^{2} d^{5} e^{2} - 180 \, A a c^{2} d^{4} e^{3} + 180 \, B a^{2} c d^{3} e^{4} - 180 \, A a^{2} c d^{2} e^{5} + 60 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7} - 2 \,{\left (7 \, B c^{3} d e^{6} - 6 \, A c^{3} e^{7}\right )} x^{6} + 3 \,{\left (7 \, B c^{3} d^{2} e^{5} - 6 \, A c^{3} d e^{6} + 15 \, B a c^{2} e^{7}\right )} x^{5} - 5 \,{\left (7 \, B c^{3} d^{3} e^{4} - 6 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 12 \, A a c^{2} e^{7}\right )} x^{4} + 10 \,{\left (7 \, B c^{3} d^{4} e^{3} - 6 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} - 30 \,{\left (7 \, B c^{3} d^{5} e^{2} - 6 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} - 6 \, A a^{2} c e^{7}\right )} x^{2} - 60 \,{\left (6 \, B c^{3} d^{6} e - 5 \, A c^{3} d^{5} e^{2} + 12 \, B a c^{2} d^{4} e^{3} - 9 \, A a c^{2} d^{3} e^{4} + 6 \, B a^{2} c d^{2} e^{5} - 3 \, A a^{2} c d e^{6}\right )} x + 60 \,{\left (7 \, B c^{3} d^{7} - 6 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} - 6 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} +{\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 )} \log \left (e x + d\right )}{60 \,{\left (e^{9} x + d 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)^2,x, algorithm="fricas")

[Out]

1/60*(10*B*c^3*e^7*x^7 + 60*B*c^3*d^7 - 60*A*c^3*d^6*e + 180*B*a*c^2*d^5*e^2 - 180*A*a*c^2*d^4*e^3 + 180*B*a^2
*c*d^3*e^4 - 180*A*a^2*c*d^2*e^5 + 60*B*a^3*d*e^6 - 60*A*a^3*e^7 - 2*(7*B*c^3*d*e^6 - 6*A*c^3*e^7)*x^6 + 3*(7*
B*c^3*d^2*e^5 - 6*A*c^3*d*e^6 + 15*B*a*c^2*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 6*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6
- 12*A*a*c^2*e^7)*x^4 + 10*(7*B*c^3*d^4*e^3 - 6*A*c^3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 + 9*B*a^
2*c*e^7)*x^3 - 30*(7*B*c^3*d^5*e^2 - 6*A*c^3*d^4*e^3 + 15*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 + 9*B*a^2*c*d*e
^6 - 6*A*a^2*c*e^7)*x^2 - 60*(6*B*c^3*d^6*e - 5*A*c^3*d^5*e^2 + 12*B*a*c^2*d^4*e^3 - 9*A*a*c^2*d^3*e^4 + 6*B*a
^2*c*d^2*e^5 - 3*A*a^2*c*d*e^6)*x + 60*(7*B*c^3*d^7 - 6*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3
+ 9*B*a^2*c*d^3*e^4 - 6*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 + (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)*log(e*x + d))/(e^9*x + d*e^8)

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Sympy [A]  time = 4.023, size = 442, normalized size = 1.43 \begin{align*} \frac{B c^{3} x^{6}}{6 e^{2}} + \frac{- A a^{3} e^{7} - 3 A a^{2} c d^{2} e^{5} - 3 A a c^{2} d^{4} e^{3} - A c^{3} d^{6} e + B a^{3} d e^{6} + 3 B a^{2} c d^{3} e^{4} + 3 B a c^{2} d^{5} e^{2} + B c^{3} d^{7}}{d e^{8} + e^{9} x} - \frac{x^{5} \left (- A c^{3} e + 2 B c^{3} d\right )}{5 e^{3}} + \frac{x^{4} \left (- 2 A c^{3} d e + 3 B a c^{2} e^{2} + 3 B c^{3} d^{2}\right )}{4 e^{4}} - \frac{x^{3} \left (- 3 A a c^{2} e^{3} - 3 A c^{3} d^{2} e + 6 B a c^{2} d e^{2} + 4 B c^{3} d^{3}\right )}{3 e^{5}} + \frac{x^{2} \left (- 6 A a c^{2} d e^{3} - 4 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 9 B a c^{2} d^{2} e^{2} + 5 B c^{3} d^{4}\right )}{2 e^{6}} - \frac{x \left (- 3 A a^{2} c e^{5} - 9 A a c^{2} d^{2} e^{3} - 5 A c^{3} d^{4} e + 6 B a^{2} c d e^{4} + 12 B a c^{2} d^{3} e^{2} + 6 B c^{3} d^{5}\right )}{e^{7}} + \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right ) \log{\left (d + e x \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)**2,x)

[Out]

B*c**3*x**6/(6*e**2) + (-A*a**3*e**7 - 3*A*a**2*c*d**2*e**5 - 3*A*a*c**2*d**4*e**3 - A*c**3*d**6*e + B*a**3*d*
e**6 + 3*B*a**2*c*d**3*e**4 + 3*B*a*c**2*d**5*e**2 + B*c**3*d**7)/(d*e**8 + e**9*x) - x**5*(-A*c**3*e + 2*B*c*
*3*d)/(5*e**3) + x**4*(-2*A*c**3*d*e + 3*B*a*c**2*e**2 + 3*B*c**3*d**2)/(4*e**4) - x**3*(-3*A*a*c**2*e**3 - 3*
A*c**3*d**2*e + 6*B*a*c**2*d*e**2 + 4*B*c**3*d**3)/(3*e**5) + x**2*(-6*A*a*c**2*d*e**3 - 4*A*c**3*d**3*e + 3*B
*a**2*c*e**4 + 9*B*a*c**2*d**2*e**2 + 5*B*c**3*d**4)/(2*e**6) - x*(-3*A*a**2*c*e**5 - 9*A*a*c**2*d**2*e**3 - 5
*A*c**3*d**4*e + 6*B*a**2*c*d*e**4 + 12*B*a*c**2*d**3*e**2 + 6*B*c**3*d**5)/e**7 + (a*e**2 + c*d**2)**2*(-6*A*
c*d*e + B*a*e**2 + 7*B*c*d**2)*log(d + e*x)/e**8

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Giac [A]  time = 1.18937, size = 728, normalized size = 2.36 \begin{align*} \frac{1}{60} \,{\left (10 \, B c^{3} - \frac{12 \,{\left (7 \, B c^{3} d e - A c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{45 \,{\left (7 \, B c^{3} d^{2} e^{2} - 2 \, A c^{3} d e^{3} + B a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{20 \,{\left (35 \, B c^{3} d^{3} e^{3} - 15 \, A c^{3} d^{2} e^{4} + 15 \, B a c^{2} d e^{5} - 3 \, A a c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{30 \,{\left (35 \, B c^{3} d^{4} e^{4} - 20 \, A c^{3} d^{3} e^{5} + 30 \, B a c^{2} d^{2} e^{6} - 12 \, A a c^{2} d e^{7} + 3 \, B a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac{180 \,{\left (7 \, B c^{3} d^{5} e^{5} - 5 \, A c^{3} d^{4} e^{6} + 10 \, B a c^{2} d^{3} e^{7} - 6 \, A a c^{2} d^{2} e^{8} + 3 \, B a^{2} c d e^{9} - A a^{2} c e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}\right )}{\left (x e + d\right )}^{6} e^{\left (-8\right )} -{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} e^{\left (-8\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B c^{3} d^{7} e^{6}}{x e + d} - \frac{A c^{3} d^{6} e^{7}}{x e + d} + \frac{3 \, B a c^{2} d^{5} e^{8}}{x e + d} - \frac{3 \, A a c^{2} d^{4} e^{9}}{x e + d} + \frac{3 \, B a^{2} c d^{3} e^{10}}{x e + d} - \frac{3 \, A a^{2} c d^{2} e^{11}}{x e + d} + \frac{B a^{3} d e^{12}}{x e + d} - \frac{A a^{3} e^{13}}{x e + d}\right )} e^{\left (-14\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)^2,x, algorithm="giac")

[Out]

1/60*(10*B*c^3 - 12*(7*B*c^3*d*e - A*c^3*e^2)*e^(-1)/(x*e + d) + 45*(7*B*c^3*d^2*e^2 - 2*A*c^3*d*e^3 + B*a*c^2
*e^4)*e^(-2)/(x*e + d)^2 - 20*(35*B*c^3*d^3*e^3 - 15*A*c^3*d^2*e^4 + 15*B*a*c^2*d*e^5 - 3*A*a*c^2*e^6)*e^(-3)/
(x*e + d)^3 + 30*(35*B*c^3*d^4*e^4 - 20*A*c^3*d^3*e^5 + 30*B*a*c^2*d^2*e^6 - 12*A*a*c^2*d*e^7 + 3*B*a^2*c*e^8)
*e^(-4)/(x*e + d)^4 - 180*(7*B*c^3*d^5*e^5 - 5*A*c^3*d^4*e^6 + 10*B*a*c^2*d^3*e^7 - 6*A*a*c^2*d^2*e^8 + 3*B*a^
2*c*d*e^9 - A*a^2*c*e^10)*e^(-5)/(x*e + d)^5)*(x*e + d)^6*e^(-8) - (7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d
^4*e^2 - 12*A*a*c^2*d^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5 + B*a^3*e^6)*e^(-8)*log(abs(x*e + d)*e^(-1)/
(x*e + d)^2) + (B*c^3*d^7*e^6/(x*e + d) - A*c^3*d^6*e^7/(x*e + d) + 3*B*a*c^2*d^5*e^8/(x*e + d) - 3*A*a*c^2*d^
4*e^9/(x*e + d) + 3*B*a^2*c*d^3*e^10/(x*e + d) - 3*A*a^2*c*d^2*e^11/(x*e + d) + B*a^3*d*e^12/(x*e + d) - A*a^3
*e^13/(x*e + d))*e^(-14)