3.24.28 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{(d+e x)^5} \, dx\) [2328]

Optimal. Leaf size=289 \[ \frac {B c^2 x}{e^5}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{4 e^6 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{3 e^6 (d+e x)^3}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{2 e^6 (d+e x)^2}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^6 (d+e x)}-\frac {c (5 B c d-2 b B e-A c e) \log (d+e x)}{e^6} \]

[Out]

B*c^2*x/e^5+1/4*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2/e^6/(e*x+d)^4+1/3*(a*e^2-b*d*e+c*d^2)*(2*A*e*(-b*e+2*c*d)-B*(
5*c*d^2-e*(-a*e+3*b*d)))/e^6/(e*x+d)^3+1/2*(B*(10*c^2*d^3+b*e^2*(-2*a*e+3*b*d)-6*c*d*e*(-a*e+2*b*d))-A*e*(6*c^
2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d)))/e^6/(e*x+d)^2+(2*A*c*e*(-b*e+2*c*d)-B*(10*c^2*d^2+b^2*e^2-2*c*e*(-a*e+4*b*d
)))/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 = 0.23, antiderivative size = 287, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \begin {gather*} \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*c^2*x)/e^5 + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(4*e^6*(d + e*x)^4) - ((c*d^2 - b*d*e + a*e^2)*(5*B*c*
d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e)))/(3*e^6*(d + e*x)^3) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) -
6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(2*e^6*(d + e*x)^2) + (2*A*c*e*(2*c
*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(e^6*(d + e*x)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*L
og[d + e*x])/e^6

Rule 785

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 (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {B c^2}{e^5}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^5}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^4}+\frac {-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^3}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\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 {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{3 e^6 (d+e x)^3}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\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+b^2 e^2-2 c e (4 b d-a e)\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 = 0.19, size = 391, normalized size = 1.35 \begin {gather*} -\frac {A e \left (-c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+e^2 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )\right )+B \left (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 )+e^2 \left (a^2 e^2 (d+4 e x)+2 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+2 c e \left (3 a e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-b d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )\right )+12 c (5 B c d-2 b B e-A c e) (d+e x)^4 \log (d+e x)}{12 e^6 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(A*e*(-(c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + e^2*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x)
+ b^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) + 2*c*e*(a*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*b*(d^3 + 4*d^2*e*x + 6*d*e^2*
x^2 + 4*e^3*x^3))) + B*(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) + e^2*(a^2*e^2*(d + 4*e*x) + 2*a*b*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*b^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 +
4*e^3*x^3)) + 2*c*e*(3*a*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - b*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*
x^2 + 48*e^3*x^3))) + 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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Maple [A]
time = 0.06, size = 444, normalized size = 1.54

method	result	size
norman	\(\frac {\frac {B \,c^{2} x^{5}}{e}-\frac {3 A \,a^{2} e^{5}+2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}-25 A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+3 B \,b^{2} d^{3} e^{2}-50 B b c \,d^{4} e +125 B \,c^{2} d^{5}}{12 e^{6}}-\frac {\left (2 A b c \,e^{2}-4 A \,c^{2} d e +2 B a c \,e^{2}+B \,b^{2} e^{2}-8 B b c d e +20 B \,c^{2} d^{2}\right ) x^{3}}{e^{3}}-\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}+6 A b c d \,e^{2}-18 A \,c^{2} d^{2} e +2 B \,e^{3} a b +6 B a c d \,e^{2}+3 B \,b^{2} d \,e^{2}-36 B b c \,d^{2} e +90 B \,c^{2} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {\left (2 A a b \,e^{4}+2 A a c d \,e^{3}+A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-22 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+2 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-44 B b c \,d^{3} e +110 B \,c^{2} d^{4}\right ) x}{3 e^{5}}}{\left (e x +d \right )^{4}}+\frac {c \left (A c e +2 b B e -5 B c d \right ) \ln \left (e x +d \right )}{e^{6}}\)	\(434\)
default	\(\frac {B \,c^{2} x}{e^{5}}+\frac {c \left (A c e +2 b B e -5 B c d \right ) \ln \left (e x +d \right )}{e^{6}}-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B \,e^{3} a b -6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{2 e^{6} \left (e x +d \right )^{2}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 B a c \,e^{2}+B \,b^{2} e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{e^{6} \left (e x +d \right )}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B b c \,d^{3} e +5 B \,c^{2} d^{4}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}}{4 e^{6} \left (e x +d \right )^{4}}\)	\(444\)
risch	\(\frac {B \,c^{2} x}{e^{5}}+\frac {\left (-2 A b c \,e^{4}+4 A \,c^{2} d \,e^{3}-2 B a c \,e^{4}-B \,b^{2} e^{4}+8 B b c d \,e^{3}-10 B \,c^{2} d^{2} e^{2}\right ) x^{3}-\frac {e \left (2 A a c \,e^{3}+A \,b^{2} e^{3}+6 A b c d \,e^{2}-18 A \,c^{2} d^{2} e +2 B \,e^{3} a b +6 B a c d \,e^{2}+3 B \,b^{2} d \,e^{2}-36 B b c \,d^{2} e +50 B \,c^{2} d^{3}\right ) x^{2}}{2}+\left (-\frac {2}{3} A a b \,e^{4}-\frac {2}{3} A a c d \,e^{3}-\frac {1}{3} A \,b^{2} d \,e^{3}-2 A b c \,d^{2} e^{2}+\frac {22}{3} A \,c^{2} d^{3} e -\frac {1}{3} B \,e^{4} a^{2}-\frac {2}{3} B a b d \,e^{3}-2 B a c \,d^{2} e^{2}-B \,b^{2} d^{2} e^{2}+\frac {44}{3} B b c \,d^{3} e -\frac {65}{3} B \,c^{2} d^{4}\right ) x -\frac {3 A \,a^{2} e^{5}+2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}-25 A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+3 B \,b^{2} d^{3} e^{2}-50 B b c \,d^{4} e +77 B \,c^{2} d^{5}}{12 e}}{e^{5} \left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right ) A}{e^{5}}+\frac {2 c \ln \left (e x +d \right ) b B}{e^{5}}-\frac {5 c^{2} \ln \left (e x +d \right ) B d}{e^{6}}\)	\(453\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 433, normalized size = 1.50 \begin {gather*} B c^{2} x e^{\left (-5\right )} - {\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} e^{\left (-6\right )} \log \left (x e + d\right ) - \frac {77 \, B c^{2} d^{5} - 25 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{4} + 3 \, {\left (B b^{2} e^{2} + 2 \, {\left (B a e^{2} + A b e^{2}\right )} c\right )} d^{3} + 12 \, {\left (10 \, B c^{2} d^{2} e^{3} + B b^{2} e^{5} + 2 \, {\left (B a e^{5} + A b e^{5}\right )} c - 4 \, {\left (2 \, B b c e^{4} + A c^{2} e^{4}\right )} d\right )} x^{3} + 3 \, A a^{2} e^{5} + {\left (2 \, B a b e^{3} + A b^{2} e^{3} + 2 \, A a c e^{3}\right )} d^{2} + 6 \, {\left (50 \, B c^{2} d^{3} e^{2} + 2 \, B a b e^{5} + A b^{2} e^{5} + 2 \, A a c e^{5} - 18 \, {\left (2 \, B b c e^{3} + A c^{2} e^{3}\right )} d^{2} + 3 \, {\left (B b^{2} e^{4} + 2 \, {\left (B a e^{4} + A b e^{4}\right )} c\right )} d\right )} x^{2} + {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} d + 4 \, {\left (65 \, B c^{2} d^{4} e - 22 \, {\left (2 \, B b c e^{2} + A c^{2} e^{2}\right )} d^{3} + B a^{2} e^{5} + 2 \, A a b e^{5} + 3 \, {\left (B b^{2} e^{3} + 2 \, {\left (B a e^{3} + A b e^{3}\right )} c\right )} d^{2} + {\left (2 \, B a b e^{4} + A b^{2} e^{4} + 2 \, A a c e^{4}\right )} d\right )} x}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.66, size = 546, normalized size = 1.89 \begin {gather*} -\frac {77 \, B c^{2} d^{5} - {\left (12 \, B c^{2} x^{5} - 12 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 3 \, A a^{2} - 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 4 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} e^{5} - {\left (48 \, B c^{2} d x^{4} + 48 \, {\left (2 \, B b c + A c^{2}\right )} d x^{3} - 18 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d x^{2} - 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d x - {\left (B a^{2} + 2 \, A a b\right )} d\right )} e^{4} + {\left (48 \, B c^{2} d^{2} x^{3} - 108 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{2} + 12 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} x + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2}\right )} e^{3} + {\left (252 \, B c^{2} d^{3} x^{2} - 88 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3}\right )} e^{2} + {\left (248 \, B c^{2} d^{4} x - 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4}\right )} e + 12 \, {\left (5 \, B c^{2} d^{5} - {\left (2 \, B b c + A c^{2}\right )} x^{4} e^{5} + {\left (5 \, B c^{2} d x^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d x^{3}\right )} e^{4} + 2 \, {\left (10 \, B c^{2} d^{2} x^{3} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{2}\right )} e^{3} + 2 \, {\left (15 \, B c^{2} d^{3} x^{2} - 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x\right )} e^{2} + {\left (20 \, B c^{2} d^{4} x - {\left (2 \, B b c + A c^{2}\right )} d^{4}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(77*B*c^2*d^5 - (12*B*c^2*x^5 - 12*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 3*A*a^2 - 6*(2*B*a*b + A*b^2 + 2*A*a*
c)*x^2 - 4*(B*a^2 + 2*A*a*b)*x)*e^5 - (48*B*c^2*d*x^4 + 48*(2*B*b*c + A*c^2)*d*x^3 - 18*(B*b^2 + 2*(B*a + A*b)
*c)*d*x^2 - 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*x - (B*a^2 + 2*A*a*b)*d)*e^4 + (48*B*c^2*d^2*x^3 - 108*(2*B*b*c +
A*c^2)*d^2*x^2 + 12*(B*b^2 + 2*(B*a + A*b)*c)*d^2*x + (2*B*a*b + A*b^2 + 2*A*a*c)*d^2)*e^3 + (252*B*c^2*d^3*x^
2 - 88*(2*B*b*c + A*c^2)*d^3*x + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3)*e^2 + (248*B*c^2*d^4*x - 25*(2*B*b*c + A*c^2
)*d^4)*e + 12*(5*B*c^2*d^5 - (2*B*b*c + A*c^2)*x^4*e^5 + (5*B*c^2*d*x^4 - 4*(2*B*b*c + A*c^2)*d*x^3)*e^4 + 2*(
10*B*c^2*d^2*x^3 - 3*(2*B*b*c + A*c^2)*d^2*x^2)*e^3 + 2*(15*B*c^2*d^3*x^2 - 2*(2*B*b*c + A*c^2)*d^3*x)*e^2 + (
20*B*c^2*d^4*x - (2*B*b*c + A*c^2)*d^4)*e)*log(x*e + d))/(x^4*e^10 + 4*d*x^3*e^9 + 6*d^2*x^2*e^8 + 4*d^3*x*e^7
+ d^4*e^6)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 719 vs. \(2 (296) = 592\).
time = 1.97, size = 719, normalized size = 2.49 \begin {gather*} {\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 \, B a c 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 \, B a c 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 \, B a c 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 \, B a c 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 {12 \, B a b e^{25}}{{\left (x e + d\right )}^{2}} + \frac {6 \, A b^{2} e^{25}}{{\left (x e + d\right )}^{2}} + \frac {12 \, A a c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {16 \, B a b d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {8 \, A b^{2} d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {16 \, A a c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {6 \, B a b d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A b^{2} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {6 \, A a c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{2} e^{26}}{{\left (x e + d\right )}^{3}} + \frac {8 \, A a b e^{26}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{2} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac {6 \, A a b d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{2} e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^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*B*a*c*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*B*a*c*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*B*a*c*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*B*a*c*d^3*e^24/(x*e + d)^4 - 6*A*b*c*d^3*e^24/(x*e + d)^4 + 12*B*a*b*e^25/(x*e + d)^2 + 6*A*b^2*e^25
/(x*e + d)^2 + 12*A*a*c*e^25/(x*e + d)^2 - 16*B*a*b*d*e^25/(x*e + d)^3 - 8*A*b^2*d*e^25/(x*e + d)^3 - 16*A*a*c
*d*e^25/(x*e + d)^3 + 6*B*a*b*d^2*e^25/(x*e + d)^4 + 3*A*b^2*d^2*e^25/(x*e + d)^4 + 6*A*a*c*d^2*e^25/(x*e + d)
^4 + 4*B*a^2*e^26/(x*e + d)^3 + 8*A*a*b*e^26/(x*e + d)^3 - 3*B*a^2*d*e^26/(x*e + d)^4 - 6*A*a*b*d*e^26/(x*e +
d)^4 + 3*A*a^2*e^27/(x*e + d)^4)*e^(-28)

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Mupad [B]
time = 2.42, size = 475, normalized size = 1.64 \begin {gather*} \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^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+2\,B\,a\,c\,e^4\right )+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+B\,a\,b\,e^4+25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2+3\,B\,a\,c\,d\,e^3+A\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{3}+\frac {2\,B\,a\,b\,d\,e^3}{3}+\frac {2\,A\,a\,b\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {2\,A\,a\,c\,d\,e^3}{3}+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 )+\frac {B\,a^2\,d\,e^4+3\,A\,a^2\,e^5+2\,B\,a\,b\,d^2\,e^3+2\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+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}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^2)/(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^3*(B*b^2*e^4 + 2*A*b*c*e^4 + 2*B*a*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) + x^2*((A*b^2*e^4)/2 + A*a*c*e^4 + B*a*b*e^4 + (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 + 3*B*a*c*d*e^3 - 18*B*b*c*d^2*e^2) + x*((B*a^2*e^4)/3 + (65*B*
c^2*d^4)/3 + (2*A*a*b*e^4)/3 + (A*b^2*d*e^3)/3 - (22*A*c^2*d^3*e)/3 + B*b^2*d^2*e^2 + (2*A*a*c*d*e^3)/3 + (2*B
*a*b*d*e^3)/3 - (44*B*b*c*d^3*e)/3 + 2*A*b*c*d^2*e^2 + 2*B*a*c*d^2*e^2) + (3*A*a^2*e^5 + 77*B*c^2*d^5 + B*a^2*
d*e^4 - 25*A*c^2*d^4*e + A*b^2*d^2*e^3 + 3*B*b^2*d^3*e^2 + 2*A*a*b*d*e^4 - 50*B*b*c*d^4*e + 2*A*a*c*d^2*e^3 +
2*B*a*b*d^2*e^3 + 6*A*b*c*d^3*e^2 + 6*B*a*c*d^3*e^2)/(12*e))/(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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