3.1123 \(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=240 \[ \frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac {c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac {B c^2 x}{e^5} \]

[Out]

B*c^2*x/e^5+1/4*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^4-1/3*d*(-b*e+c*d)*(B*d*(-3*b*e+5*c*d)-2*A*e*(-b*e+2*c
*d))/e^6/(e*x+d)^3+1/2*(-A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)+B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))/e^6/(e*x+d)^
2+(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))/e^6/(e*x+d)-c*(-A*c*e-2*B*b*e+5*B*c*d)*ln(e*x+d)/e^6

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Rubi [A]  time = 0.28, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \[ \frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac {c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac {B c^2 x}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

(B*c^2*x)/e^5 + (d^2*(B*d - A*e)*(c*d - b*e)^2)/(4*e^6*(d + e*x)^4) - (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*
A*e*(2*c*d - b*e)))/(3*e^6*(d + e*x)^3) - (A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))/(2*e^6*(d + e*x)^2) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))/(e^6*(d +
e*x)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*Log[d + e*x])/e^6

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {B c^2}{e^5}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^4}+\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)^3}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^5 (d+e x)^2}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {B c^2 x}{e^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{2 e^6 (d+e x)^2}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^6 (d+e x)}-\frac {c (5 B c d-2 b B e-A c e) \log (d+e x)}{e^6}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 275, normalized size = 1.15 \[ -\frac {A e \left (b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 b c e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+c^2 (-d) \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) (-A c e-2 b B e+5 B c d)+B \left (3 b^2 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-2 b c d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

-1/12*(A*e*(b^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*b*c*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - c^2*d*
(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + B*(3*b^2*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)
- 2*b*c*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + c^2*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2
+ 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + 12*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^4*Log[d + e*x])/(e
^6*(d + e*x)^4)

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fricas [B]  time = 0.61, size = 475, normalized size = 1.98 \[ \frac {12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} + 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 12 \, {\left (4 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 6 \, {\left (42 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 4 \, {\left (62 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 12 \, {\left (5 \, B c^{2} d^{5} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B c^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B c^{2} d^{4} e - {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(12*B*c^2*e^5*x^5 + 48*B*c^2*d*e^4*x^4 - 77*B*c^2*d^5 - A*b^2*d^2*e^3 + 25*(2*B*b*c + A*c^2)*d^4*e - 3*(B
*b^2 + 2*A*b*c)*d^3*e^2 - 12*(4*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*A*b*c)*e^5)*x^3 - 6*(42
*B*c^2*d^3*e^2 + A*b^2*e^5 - 18*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 - 4*(62*B*c^2*d^4*e
+ A*b^2*d*e^4 - 22*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x - 12*(5*B*c^2*d^5 - (2*B*b*c +
A*c^2)*d^4*e + (5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 4*(5*B*c^2*d^2*e^3 - (2*B*b*c + A*c^2)*d*e^4)*x^3
+ 6*(5*B*c^2*d^3*e^2 - (2*B*b*c + A*c^2)*d^2*e^3)*x^2 + 4*(5*B*c^2*d^4*e - (2*B*b*c + A*c^2)*d^3*e^2)*x)*log(
e*x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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giac [B]  time = 0.18, size = 474, normalized size = 1.98 \[ {\left (x e + d\right )} B c^{2} e^{\left (-6\right )} + {\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac {60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {96 \, B b c d e^{23}}{x e + d} - \frac {48 \, A c^{2} d e^{23}}{x e + d} + \frac {72 \, B b c d^{2} e^{23}}{{\left (x e + d\right )}^{2}} + \frac {36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {32 \, B b c d^{3} e^{23}}{{\left (x e + d\right )}^{3}} - \frac {16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {6 \, B b c d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {12 \, B b^{2} e^{24}}{x e + d} + \frac {24 \, A b c e^{24}}{x e + d} - \frac {18 \, B b^{2} d e^{24}}{{\left (x e + d\right )}^{2}} - \frac {36 \, A b c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {12 \, B b^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac {24 \, A b c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B b^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac {6 \, A b c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {6 \, A b^{2} e^{25}}{{\left (x e + d\right )}^{2}} - \frac {8 \, A b^{2} d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {3 \, A b^{2} d^{2} e^{25}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

(x*e + d)*B*c^2*e^(-6) + (5*B*c^2*d - 2*B*b*c*e - A*c^2*e)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*
(120*B*c^2*d^2*e^22/(x*e + d) - 60*B*c^2*d^3*e^22/(x*e + d)^2 + 20*B*c^2*d^4*e^22/(x*e + d)^3 - 3*B*c^2*d^5*e^
22/(x*e + d)^4 - 96*B*b*c*d*e^23/(x*e + d) - 48*A*c^2*d*e^23/(x*e + d) + 72*B*b*c*d^2*e^23/(x*e + d)^2 + 36*A*
c^2*d^2*e^23/(x*e + d)^2 - 32*B*b*c*d^3*e^23/(x*e + d)^3 - 16*A*c^2*d^3*e^23/(x*e + d)^3 + 6*B*b*c*d^4*e^23/(x
*e + d)^4 + 3*A*c^2*d^4*e^23/(x*e + d)^4 + 12*B*b^2*e^24/(x*e + d) + 24*A*b*c*e^24/(x*e + d) - 18*B*b^2*d*e^24
/(x*e + d)^2 - 36*A*b*c*d*e^24/(x*e + d)^2 + 12*B*b^2*d^2*e^24/(x*e + d)^3 + 24*A*b*c*d^2*e^24/(x*e + d)^3 - 3
*B*b^2*d^3*e^24/(x*e + d)^4 - 6*A*b*c*d^3*e^24/(x*e + d)^4 + 6*A*b^2*e^25/(x*e + d)^2 - 8*A*b^2*d*e^25/(x*e +
d)^3 + 3*A*b^2*d^2*e^25/(x*e + d)^4)*e^(-28)

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maple [A]  time = 0.06, size = 465, normalized size = 1.94 \[ -\frac {A \,b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {A b c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {A \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {B \,b^{2} d^{3}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {B b c \,d^{4}}{2 \left (e x +d \right )^{4} e^{5}}+\frac {B \,c^{2} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}+\frac {2 A \,b^{2} d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 A b c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 A \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {B \,b^{2} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {8 B b c \,d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {5 B \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A \,b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 A b c d}{\left (e x +d \right )^{2} e^{4}}-\frac {3 A \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {3 B \,b^{2} d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {6 B b c \,d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {5 B \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{6}}-\frac {2 A b c}{\left (e x +d \right ) e^{4}}+\frac {4 A \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {A \,c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {B \,b^{2}}{\left (e x +d \right ) e^{4}}+\frac {8 B b c d}{\left (e x +d \right ) e^{5}}+\frac {2 B b c \ln \left (e x +d \right )}{e^{5}}-\frac {10 B \,c^{2} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {5 B \,c^{2} d \ln \left (e x +d \right )}{e^{6}}+\frac {B \,c^{2} x}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^5,x)

[Out]

-1/e^4/(e*x+d)*B*b^2+c^2/e^5*ln(e*x+d)*A-1/2/e^3/(e*x+d)^2*A*b^2+2*c/e^5*ln(e*x+d)*B*b-5*c^2/e^6*ln(e*x+d)*B*d
-d^2/e^4/(e*x+d)^3*B*b^2-5/3*d^4/e^6/(e*x+d)^3*B*c^2-2/e^4/(e*x+d)*A*b*c+4/e^5/(e*x+d)*A*c^2*d-1/2*d^4/e^5/(e*
x+d)^4*B*b*c+8/e^5/(e*x+d)*B*b*c*d+1/2*d^3/e^4/(e*x+d)^4*A*b*c+3/e^4/(e*x+d)^2*A*b*c*d-6/e^5/(e*x+d)^2*B*b*c*d
^2+8/3*d^3/e^5/(e*x+d)^3*B*b*c-2*d^2/e^4/(e*x+d)^3*A*b*c+5/e^6/(e*x+d)^2*B*c^2*d^3+2/3*d/e^3/(e*x+d)^3*A*b^2+4
/3*d^3/e^5/(e*x+d)^3*A*c^2-10/e^6/(e*x+d)*B*c^2*d^2+1/4*d^3/e^4/(e*x+d)^4*B*b^2+1/4*d^5/e^6/(e*x+d)^4*B*c^2-3/
e^5/(e*x+d)^2*A*c^2*d^2+3/2/e^4/(e*x+d)^2*B*b^2*d-1/4*d^2/e^3/(e*x+d)^4*A*b^2-1/4*d^4/e^5/(e*x+d)^4*A*c^2+B*c^
2*x/e^5

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maxima [A]  time = 0.58, size = 321, normalized size = 1.34 \[ -\frac {77 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} - 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 12 \, {\left (10 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 6 \, {\left (50 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (65 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac {B c^{2} x}{e^{5}} - \frac {{\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(77*B*c^2*d^5 + A*b^2*d^2*e^3 - 25*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 12*(10*B*c^2*
d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*A*b*c)*e^5)*x^3 + 6*(50*B*c^2*d^3*e^2 + A*b^2*e^5 - 18*(2*B*b
*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(65*B*c^2*d^4*e + A*b^2*d*e^4 - 22*(2*B*b*c + A*c^2)*
d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + B
*c^2*x/e^5 - (5*B*c^2*d - (2*B*b*c + A*c^2)*e)*log(e*x + d)/e^6

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mupad [B]  time = 1.47, size = 338, normalized size = 1.41 \[ \frac {\ln \left (d+e\,x\right )\,\left (A\,c^2\,e-5\,B\,c^2\,d+2\,B\,b\,c\,e\right )}{e^6}-\frac {x^2\,\left (\frac {3\,B\,b^2\,d\,e^3}{2}+\frac {A\,b^2\,e^4}{2}-18\,B\,b\,c\,d^2\,e^2+3\,A\,b\,c\,d\,e^3+25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2\right )+\frac {3\,B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3-50\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2+77\,B\,c^2\,d^5-25\,A\,c^2\,d^4\,e}{12\,e}+x\,\left (B\,b^2\,d^2\,e^2+\frac {A\,b^2\,d\,e^3}{3}-\frac {44\,B\,b\,c\,d^3\,e}{3}+2\,A\,b\,c\,d^2\,e^2+\frac {65\,B\,c^2\,d^4}{3}-\frac {22\,A\,c^2\,d^3\,e}{3}\right )+x^3\,\left (B\,b^2\,e^4-8\,B\,b\,c\,d\,e^3+2\,A\,b\,c\,e^4+10\,B\,c^2\,d^2\,e^2-4\,A\,c^2\,d\,e^3\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,c^2\,x}{e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^2*(A + B*x))/(d + e*x)^5,x)

[Out]

(log(d + e*x)*(A*c^2*e - 5*B*c^2*d + 2*B*b*c*e))/e^6 - (x^2*((A*b^2*e^4)/2 + (3*B*b^2*d*e^3)/2 + 25*B*c^2*d^3*
e - 9*A*c^2*d^2*e^2 + 3*A*b*c*d*e^3 - 18*B*b*c*d^2*e^2) + (77*B*c^2*d^5 - 25*A*c^2*d^4*e + A*b^2*d^2*e^3 + 3*B
*b^2*d^3*e^2 - 50*B*b*c*d^4*e + 6*A*b*c*d^3*e^2)/(12*e) + x*((65*B*c^2*d^4)/3 + (A*b^2*d*e^3)/3 - (22*A*c^2*d^
3*e)/3 + B*b^2*d^2*e^2 - (44*B*b*c*d^3*e)/3 + 2*A*b*c*d^2*e^2) + x^3*(B*b^2*e^4 + 2*A*b*c*e^4 - 4*A*c^2*d*e^3
+ 10*B*c^2*d^2*e^2 - 8*B*b*c*d*e^3))/(d^4*e^5 + e^9*x^4 + 4*d^3*e^6*x + 4*d*e^8*x^3 + 6*d^2*e^7*x^2) + (B*c^2*
x)/e^5

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sympy [A]  time = 32.26, size = 381, normalized size = 1.59 \[ \frac {B c^{2} x}{e^{5}} + \frac {c \left (A c e + 2 B b e - 5 B c d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- A b^{2} d^{2} e^{3} - 6 A b c d^{3} e^{2} + 25 A c^{2} d^{4} e - 3 B b^{2} d^{3} e^{2} + 50 B b c d^{4} e - 77 B c^{2} d^{5} + x^{3} \left (- 24 A b c e^{5} + 48 A c^{2} d e^{4} - 12 B b^{2} e^{5} + 96 B b c d e^{4} - 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 6 A b^{2} e^{5} - 36 A b c d e^{4} + 108 A c^{2} d^{2} e^{3} - 18 B b^{2} d e^{4} + 216 B b c d^{2} e^{3} - 300 B c^{2} d^{3} e^{2}\right ) + x \left (- 4 A b^{2} d e^{4} - 24 A b c d^{2} e^{3} + 88 A c^{2} d^{3} e^{2} - 12 B b^{2} d^{2} e^{3} + 176 B b c d^{3} e^{2} - 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**5,x)

[Out]

B*c**2*x/e**5 + c*(A*c*e + 2*B*b*e - 5*B*c*d)*log(d + e*x)/e**6 + (-A*b**2*d**2*e**3 - 6*A*b*c*d**3*e**2 + 25*
A*c**2*d**4*e - 3*B*b**2*d**3*e**2 + 50*B*b*c*d**4*e - 77*B*c**2*d**5 + x**3*(-24*A*b*c*e**5 + 48*A*c**2*d*e**
4 - 12*B*b**2*e**5 + 96*B*b*c*d*e**4 - 120*B*c**2*d**2*e**3) + x**2*(-6*A*b**2*e**5 - 36*A*b*c*d*e**4 + 108*A*
c**2*d**2*e**3 - 18*B*b**2*d*e**4 + 216*B*b*c*d**2*e**3 - 300*B*c**2*d**3*e**2) + x*(-4*A*b**2*d*e**4 - 24*A*b
*c*d**2*e**3 + 88*A*c**2*d**3*e**2 - 12*B*b**2*d**2*e**3 + 176*B*b*c*d**3*e**2 - 260*B*c**2*d**4*e))/(12*d**4*
e**6 + 48*d**3*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e**10*x**4)

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