3.1319 \(\int \frac {(A+B x) (a+c x^2)^3}{d+e x} \, dx\)

Optimal. Leaf size=290 \[ \frac {x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}-\frac {c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}-\frac {c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac {c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}+\frac {c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac {\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac {c^3 x^6 (B d-A e)}{6 e^2}+\frac {B c^3 x^7}{7 e} \]

[Out]

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

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Rubi [A]  time = 0.40, antiderivative size = 290, 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} \[ -\frac {c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac {c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}+\frac {x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}+\frac {c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac {c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac {\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac {c^3 x^6 (B d-A e)}{6 e^2}+\frac {B c^3 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.18, size = 311, normalized size = 1.07 \[ \frac {e x \left (7 A c e \left (90 a^2 e^4 (e x-2 d)+15 a c e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+B \left (420 a^3 e^6+210 a^2 c e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+21 a c^2 e^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+c^3 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 \left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{420 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(7*A*c*e*(90*a^2*e^4*(-2*d + e*x) + 15*a*c*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + c^2*(-60
*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + B*(420*a^3*e^6 + 210*a^2*c
*e^4*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 21*a*c^2*e^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^
4*x^4) + c^3*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e
^6*x^6))) - 420*(B*d - A*e)*(c*d^2 + a*e^2)^3*Log[d + e*x])/(420*e^8)

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fricas [A]  time = 0.58, size = 448, normalized size = 1.54 \[ \frac {60 \, B c^{3} e^{7} x^{7} - 70 \, {\left (B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{5} - A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5} + 3 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4} + 3 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3} + 3 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e - A c^{3} d^{5} e^{2} + 3 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} - 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 420 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d),x, algorithm="fricas")

[Out]

1/420*(60*B*c^3*e^7*x^7 - 70*(B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 84*(B*c^3*d^2*e^5 - A*c^3*d*e^6 + 3*B*a*c^2*e^7)*
x^5 - 105*(B*c^3*d^3*e^4 - A*c^3*d^2*e^5 + 3*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 140*(B*c^3*d^4*e^3 - A*c^3*d
^3*e^4 + 3*B*a*c^2*d^2*e^5 - 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 - 210*(B*c^3*d^5*e^2 - A*c^3*d^4*e^3 + 3*B*a
*c^2*d^3*e^4 - 3*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 - 3*A*a^2*c*e^7)*x^2 + 420*(B*c^3*d^6*e - 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 + 3*B*a^2*c*d^2*e^5 - 3*A*a^2*c*d*e^6 + B*a^3*e^7)*x - 420*(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)*log(e*x + d))/e^8

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giac [A]  time = 0.18, size = 459, normalized size = 1.58 \[ -{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (60 \, B c^{3} x^{7} e^{6} - 70 \, B c^{3} d x^{6} e^{5} + 84 \, B c^{3} d^{2} x^{5} e^{4} - 105 \, B c^{3} d^{3} x^{4} e^{3} + 140 \, B c^{3} d^{4} x^{3} e^{2} - 210 \, B c^{3} d^{5} x^{2} e + 420 \, B c^{3} d^{6} x + 70 \, A c^{3} x^{6} e^{6} - 84 \, A c^{3} d x^{5} e^{5} + 105 \, A c^{3} d^{2} x^{4} e^{4} - 140 \, A c^{3} d^{3} x^{3} e^{3} + 210 \, A c^{3} d^{4} x^{2} e^{2} - 420 \, A c^{3} d^{5} x e + 252 \, B a c^{2} x^{5} e^{6} - 315 \, B a c^{2} d x^{4} e^{5} + 420 \, B a c^{2} d^{2} x^{3} e^{4} - 630 \, B a c^{2} d^{3} x^{2} e^{3} + 1260 \, B a c^{2} d^{4} x e^{2} + 315 \, A a c^{2} x^{4} e^{6} - 420 \, A a c^{2} d x^{3} e^{5} + 630 \, A a c^{2} d^{2} x^{2} e^{4} - 1260 \, A a c^{2} d^{3} x e^{3} + 420 \, B a^{2} c x^{3} e^{6} - 630 \, B a^{2} c d x^{2} e^{5} + 1260 \, B a^{2} c d^{2} x e^{4} + 630 \, A a^{2} c x^{2} e^{6} - 1260 \, A a^{2} c d x e^{5} + 420 \, B a^{3} x e^{6}\right )} e^{\left (-7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d),x, algorithm="giac")

[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^(-8)*log(abs(x*e + d)) + 1/420*(60*B*c^3*x^7*e^6 - 70*B*c^3*d*x^6*e^5 + 84*B*c^3*d^2*
x^5*e^4 - 105*B*c^3*d^3*x^4*e^3 + 140*B*c^3*d^4*x^3*e^2 - 210*B*c^3*d^5*x^2*e + 420*B*c^3*d^6*x + 70*A*c^3*x^6
*e^6 - 84*A*c^3*d*x^5*e^5 + 105*A*c^3*d^2*x^4*e^4 - 140*A*c^3*d^3*x^3*e^3 + 210*A*c^3*d^4*x^2*e^2 - 420*A*c^3*
d^5*x*e + 252*B*a*c^2*x^5*e^6 - 315*B*a*c^2*d*x^4*e^5 + 420*B*a*c^2*d^2*x^3*e^4 - 630*B*a*c^2*d^3*x^2*e^3 + 12
60*B*a*c^2*d^4*x*e^2 + 315*A*a*c^2*x^4*e^6 - 420*A*a*c^2*d*x^3*e^5 + 630*A*a*c^2*d^2*x^2*e^4 - 1260*A*a*c^2*d^
3*x*e^3 + 420*B*a^2*c*x^3*e^6 - 630*B*a^2*c*d*x^2*e^5 + 1260*B*a^2*c*d^2*x*e^4 + 630*A*a^2*c*x^2*e^6 - 1260*A*
a^2*c*d*x*e^5 + 420*B*a^3*x*e^6)*e^(-7)

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maple [A]  time = 0.05, size = 526, normalized size = 1.81 \[ \frac {B \,c^{3} x^{7}}{7 e}+\frac {A \,c^{3} x^{6}}{6 e}-\frac {B \,c^{3} d \,x^{6}}{6 e^{2}}-\frac {A \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 B a \,c^{2} x^{5}}{5 e}+\frac {B \,c^{3} d^{2} x^{5}}{5 e^{3}}+\frac {3 A a \,c^{2} x^{4}}{4 e}+\frac {A \,c^{3} d^{2} x^{4}}{4 e^{3}}-\frac {3 B a \,c^{2} d \,x^{4}}{4 e^{2}}-\frac {B \,c^{3} d^{3} x^{4}}{4 e^{4}}-\frac {A a \,c^{2} d \,x^{3}}{e^{2}}-\frac {A \,c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {B \,a^{2} c \,x^{3}}{e}+\frac {B a \,c^{2} d^{2} x^{3}}{e^{3}}+\frac {B \,c^{3} d^{4} x^{3}}{3 e^{5}}+\frac {3 A \,a^{2} c \,x^{2}}{2 e}+\frac {3 A a \,c^{2} d^{2} x^{2}}{2 e^{3}}+\frac {A \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {3 B \,a^{2} c d \,x^{2}}{2 e^{2}}-\frac {3 B a \,c^{2} d^{3} x^{2}}{2 e^{4}}-\frac {B \,c^{3} d^{5} x^{2}}{2 e^{6}}+\frac {A \,a^{3} \ln \left (e x +d \right )}{e}+\frac {3 A \,a^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 A \,a^{2} c d x}{e^{2}}+\frac {3 A a \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {3 A a \,c^{2} d^{3} x}{e^{4}}+\frac {A \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {A \,c^{3} d^{5} x}{e^{6}}-\frac {B \,a^{3} d \ln \left (e x +d \right )}{e^{2}}+\frac {B \,a^{3} x}{e}-\frac {3 B \,a^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {3 B \,a^{2} c \,d^{2} x}{e^{3}}-\frac {3 B a \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {3 B a \,c^{2} d^{4} x}{e^{5}}-\frac {B \,c^{3} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {B \,c^{3} d^{6} x}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 447, normalized size = 1.54 \[ \frac {60 \, B c^{3} e^{6} x^{7} - 70 \, {\left (B c^{3} d e^{5} - A c^{3} e^{6}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{4} - A c^{3} d e^{5} + 3 \, B a c^{2} e^{6}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{3} - A c^{3} d^{2} e^{4} + 3 \, B a c^{2} d e^{5} - 3 \, A a c^{2} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} - A c^{3} d^{3} e^{3} + 3 \, B a c^{2} d^{2} e^{4} - 3 \, A a c^{2} d e^{5} + 3 \, B a^{2} c e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e - A c^{3} d^{4} e^{2} + 3 \, B a c^{2} d^{3} e^{3} - 3 \, A a c^{2} d^{2} e^{4} + 3 \, B a^{2} c d e^{5} - 3 \, A a^{2} c e^{6}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} - A c^{3} d^{5} e + 3 \, B a c^{2} d^{4} e^{2} - 3 \, A a c^{2} d^{3} e^{3} + 3 \, B a^{2} c d^{2} e^{4} - 3 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d),x, algorithm="maxima")

[Out]

1/420*(60*B*c^3*e^6*x^7 - 70*(B*c^3*d*e^5 - A*c^3*e^6)*x^6 + 84*(B*c^3*d^2*e^4 - A*c^3*d*e^5 + 3*B*a*c^2*e^6)*
x^5 - 105*(B*c^3*d^3*e^3 - A*c^3*d^2*e^4 + 3*B*a*c^2*d*e^5 - 3*A*a*c^2*e^6)*x^4 + 140*(B*c^3*d^4*e^2 - A*c^3*d
^3*e^3 + 3*B*a*c^2*d^2*e^4 - 3*A*a*c^2*d*e^5 + 3*B*a^2*c*e^6)*x^3 - 210*(B*c^3*d^5*e - A*c^3*d^4*e^2 + 3*B*a*c
^2*d^3*e^3 - 3*A*a*c^2*d^2*e^4 + 3*B*a^2*c*d*e^5 - 3*A*a^2*c*e^6)*x^2 + 420*(B*c^3*d^6 - A*c^3*d^5*e + 3*B*a*c
^2*d^4*e^2 - 3*A*a*c^2*d^3*e^3 + 3*B*a^2*c*d^2*e^4 - 3*A*a^2*c*d*e^5 + B*a^3*e^6)*x)/e^7 - (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)*log(e*x + d)/e^8

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mupad [B]  time = 0.08, size = 494, normalized size = 1.70 \[ x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{2\,e}+\frac {3\,A\,a^2\,c}{2\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{4\,e}+\frac {3\,A\,a\,c^2}{4\,e}\right )+x^6\,\left (\frac {A\,c^3}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{3\,e}-\frac {B\,a^2\,c}{e}\right )+x\,\left (\frac {B\,a^3}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{e}+\frac {3\,A\,a^2\,c}{e}\right )}{e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}-\frac {3\,B\,a\,c^2}{5\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^3\,d\,e^6+A\,a^3\,e^7-3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5-3\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3-B\,c^3\,d^7+A\,c^3\,d^6\,e\right )}{e^8}+\frac {B\,c^3\,x^7}{7\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.92, size = 410, normalized size = 1.41 \[ \frac {B c^{3} x^{7}}{7 e} + x^{6} \left (\frac {A c^{3}}{6 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \left (- \frac {A c^{3} d}{5 e^{2}} + \frac {3 B a c^{2}}{5 e} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {3 A a c^{2}}{4 e} + \frac {A c^{3} d^{2}}{4 e^{3}} - \frac {3 B a c^{2} d}{4 e^{2}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \left (- \frac {A a c^{2} d}{e^{2}} - \frac {A c^{3} d^{3}}{3 e^{4}} + \frac {B a^{2} c}{e} + \frac {B a c^{2} d^{2}}{e^{3}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \left (\frac {3 A a^{2} c}{2 e} + \frac {3 A a c^{2} d^{2}}{2 e^{3}} + \frac {A c^{3} d^{4}}{2 e^{5}} - \frac {3 B a^{2} c d}{2 e^{2}} - \frac {3 B a c^{2} d^{3}}{2 e^{4}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (- \frac {3 A a^{2} c d}{e^{2}} - \frac {3 A a c^{2} d^{3}}{e^{4}} - \frac {A c^{3} d^{5}}{e^{6}} + \frac {B a^{3}}{e} + \frac {3 B a^{2} c d^{2}}{e^{3}} + \frac {3 B a c^{2} d^{4}}{e^{5}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d),x)

[Out]

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

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