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

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

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Rubi [A]  time = 0.44, antiderivative size = 309, 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 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} \end {gather*}

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*}

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Mathematica [A]  time = 0.18, size = 405, normalized size = 1.31 \begin {gather*} \frac {6 A e \left (-10 a^3 e^6+30 a^2 c e^4 \left (-d^2+d e x+e^2 x^2\right )+10 a c^2 e^2 \left (-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\right )+c^3 \left (-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\right )\right )+B \left (60 a^3 d e^6+90 a^2 c e^4 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+15 a c^2 e^2 \left (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\right )+c^3 \left (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\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)} \end {gather*}

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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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)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 621, normalized size = 2.01 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.20, size = 539, normalized size = 1.74 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.06, size = 558, normalized size = 1.81 \begin {gather*} \frac {B \,c^{3} x^{6}}{6 e^{2}}+\frac {A \,c^{3} x^{5}}{5 e^{2}}-\frac {2 B \,c^{3} d \,x^{5}}{5 e^{3}}-\frac {A \,c^{3} d \,x^{4}}{2 e^{3}}+\frac {3 B a \,c^{2} x^{4}}{4 e^{2}}+\frac {3 B \,c^{3} d^{2} x^{4}}{4 e^{4}}+\frac {A a \,c^{2} x^{3}}{e^{2}}+\frac {A \,c^{3} d^{2} x^{3}}{e^{4}}-\frac {2 B a \,c^{2} d \,x^{3}}{e^{3}}-\frac {4 B \,c^{3} d^{3} x^{3}}{3 e^{5}}-\frac {3 A a \,c^{2} d \,x^{2}}{e^{3}}-\frac {2 A \,c^{3} d^{3} x^{2}}{e^{5}}+\frac {3 B \,a^{2} c \,x^{2}}{2 e^{2}}+\frac {9 B a \,c^{2} d^{2} x^{2}}{2 e^{4}}+\frac {5 B \,c^{3} d^{4} x^{2}}{2 e^{6}}-\frac {A \,a^{3}}{\left (e x +d \right ) e}-\frac {3 A \,a^{2} c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {6 A \,a^{2} c d \ln \left (e x +d \right )}{e^{3}}+\frac {3 A \,a^{2} c x}{e^{2}}-\frac {3 A a \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {12 A a \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {9 A a \,c^{2} d^{2} x}{e^{4}}-\frac {A \,c^{3} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {6 A \,c^{3} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {5 A \,c^{3} d^{4} x}{e^{6}}+\frac {B \,a^{3} d}{\left (e x +d \right ) e^{2}}+\frac {B \,a^{3} \ln \left (e x +d \right )}{e^{2}}+\frac {3 B \,a^{2} c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {9 B \,a^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {6 B \,a^{2} c d x}{e^{3}}+\frac {3 B a \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {15 B a \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {12 B a \,c^{2} d^{3} x}{e^{5}}+\frac {B \,c^{3} d^{7}}{\left (e x +d \right ) e^{8}}+\frac {7 B \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{8}}-\frac {6 B \,c^{3} d^{5} 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)^2,x)

[Out]

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

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maxima [A]  time = 0.56, size = 456, normalized size = 1.48 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.74, size = 826, normalized size = 2.67 \begin {gather*} x^3\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e}-\frac {3\,B\,a\,c^2}{e^2}+\frac {B\,c^3\,d^2}{e^4}\right )}{3\,e}-\frac {d^2\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{3\,e^2}+\frac {A\,a\,c^2}{e^2}\right )+x^5\,\left (\frac {A\,c^3}{5\,e^2}-\frac {2\,B\,c^3\,d}{5\,e^3}\right )-x^4\,\left (\frac {d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{2\,e}-\frac {3\,B\,a\,c^2}{4\,e^2}+\frac {B\,c^3\,d^2}{4\,e^4}\right )-x\,\left (\frac {2\,d\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e}-\frac {3\,B\,a\,c^2}{e^2}+\frac {B\,c^3\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e}-\frac {3\,B\,a\,c^2}{e^2}+\frac {B\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^2}\right )}{e}+\frac {3\,B\,a^2\,c}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e}-\frac {3\,B\,a\,c^2}{e^2}+\frac {B\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^2}\right )}{e^2}-\frac {3\,A\,a^2\,c}{e^2}\right )+x^2\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e}-\frac {3\,B\,a\,c^2}{e^2}+\frac {B\,c^3\,d^2}{e^4}\right )}{2\,e^2}-\frac {d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e}-\frac {3\,B\,a\,c^2}{e^2}+\frac {B\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {A\,c^3}{e^2}-\frac {2\,B\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^2}\right )}{e}+\frac {3\,B\,a^2\,c}{2\,e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (B\,a^3\,e^6+9\,B\,a^2\,c\,d^2\,e^4-6\,A\,a^2\,c\,d\,e^5+15\,B\,a\,c^2\,d^4\,e^2-12\,A\,a\,c^2\,d^3\,e^3+7\,B\,c^3\,d^6-6\,A\,c^3\,d^5\,e\right )}{e^8}-\frac {-B\,a^3\,d\,e^6+A\,a^3\,e^7-3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5-3\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3-B\,c^3\,d^7+A\,c^3\,d^6\,e}{e\,\left (x\,e^8+d\,e^7\right )}+\frac {B\,c^3\,x^6}{6\,e^2} \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)^2,x)

[Out]

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

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sympy [A]  time = 1.96, size = 454, normalized size = 1.47 \begin {gather*} \frac {B c^{3} x^{6}}{6 e^{2}} + x^{5} \left (\frac {A c^{3}}{5 e^{2}} - \frac {2 B c^{3} d}{5 e^{3}}\right ) + x^{4} \left (- \frac {A c^{3} d}{2 e^{3}} + \frac {3 B a c^{2}}{4 e^{2}} + \frac {3 B c^{3} d^{2}}{4 e^{4}}\right ) + x^{3} \left (\frac {A a c^{2}}{e^{2}} + \frac {A c^{3} d^{2}}{e^{4}} - \frac {2 B a c^{2} d}{e^{3}} - \frac {4 B c^{3} d^{3}}{3 e^{5}}\right ) + x^{2} \left (- \frac {3 A a c^{2} d}{e^{3}} - \frac {2 A c^{3} d^{3}}{e^{5}} + \frac {3 B a^{2} c}{2 e^{2}} + \frac {9 B a c^{2} d^{2}}{2 e^{4}} + \frac {5 B c^{3} d^{4}}{2 e^{6}}\right ) + x \left (\frac {3 A a^{2} c}{e^{2}} + \frac {9 A a c^{2} d^{2}}{e^{4}} + \frac {5 A c^{3} d^{4}}{e^{6}} - \frac {6 B a^{2} c d}{e^{3}} - \frac {12 B a c^{2} d^{3}}{e^{5}} - \frac {6 B c^{3} d^{5}}{e^{7}}\right ) + \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 {\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 {gather*}

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