3.12.55 \(\int \frac {(A+B x) (d+e x)^4}{(b x+c x^2)^3} \, dx\) [1155]

Optimal. Leaf size=235 \[ -\frac {A d^4}{2 b^3 x^2}-\frac {d^3 (b B d-3 A c d+4 A b e)}{b^4 x}-\frac {(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac {(c d-b e)^3 \left (2 b B c d-3 A c^2 d+2 b^2 B e-A b c e\right )}{b^4 c^3 (b+c x)}+\frac {d^2 \left (6 A c^2 d^2+2 b^2 e (2 B d+3 A e)-3 b c d (B d+4 A e)\right ) \log (x)}{b^5}+\frac {(c d-b e)^2 \left (3 b B c^2 d^2-6 A c^3 d^2+2 b^2 B c d e+b^3 B e^2\right ) \log (b+c x)}{b^5 c^3} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 235, 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 = {785} \begin {gather*} -\frac {d^3 (4 A b e-3 A c d+b B d)}{b^4 x}-\frac {(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac {A d^4}{2 b^3 x^2}+\frac {d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac {(c d-b e)^3 \left (-A b c e-3 A c^2 d+2 b^2 B e+2 b B c d\right )}{b^4 c^3 (b+c x)}+\frac {(c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+b^3 B e^2+2 b^2 B c d e+3 b B c^2 d^2\right )}{b^5 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^3,x]

[Out]

-1/2*(A*d^4)/(b^3*x^2) - (d^3*(b*B*d - 3*A*c*d + 4*A*b*e))/(b^4*x) - ((b*B - A*c)*(c*d - b*e)^4)/(2*b^3*c^3*(b
+ c*x)^2) - ((c*d - b*e)^3*(2*b*B*c*d - 3*A*c^2*d + 2*b^2*B*e - A*b*c*e))/(b^4*c^3*(b + c*x)) + (d^2*(6*A*c^2
*d^2 + 2*b^2*e*(2*B*d + 3*A*e) - 3*b*c*d*(B*d + 4*A*e))*Log[x])/b^5 + ((c*d - b*e)^2*(3*b*B*c^2*d^2 - 6*A*c^3*
d^2 + 2*b^2*B*c*d*e + b^3*B*e^2)*Log[b + c*x])/(b^5*c^3)

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

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Mathematica [A]
time = 0.12, size = 228, normalized size = 0.97 \begin {gather*} -\frac {\frac {A b^2 d^4}{x^2}+\frac {2 b d^3 (b B d-3 A c d+4 A b e)}{x}+\frac {b^2 (b B-A c) (c d-b e)^4}{c^3 (b+c x)^2}-\frac {2 b (-c d+b e)^3 \left (-3 A c^2 d+2 b^2 B e+b c (2 B d-A e)\right )}{c^3 (b+c x)}-2 d^2 \left (6 A c^2 d^2+2 b^2 e (2 B d+3 A e)-3 b c d (B d+4 A e)\right ) \log (x)-\frac {2 (c d-b e)^2 \left (3 b B c^2 d^2-6 A c^3 d^2+2 b^2 B c d e+b^3 B e^2\right ) \log (b+c x)}{c^3}}{2 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^3,x]

[Out]

-1/2*((A*b^2*d^4)/x^2 + (2*b*d^3*(b*B*d - 3*A*c*d + 4*A*b*e))/x + (b^2*(b*B - A*c)*(c*d - b*e)^4)/(c^3*(b + c*
x)^2) - (2*b*(-(c*d) + b*e)^3*(-3*A*c^2*d + 2*b^2*B*e + b*c*(2*B*d - A*e)))/(c^3*(b + c*x)) - 2*d^2*(6*A*c^2*d
^2 + 2*b^2*e*(2*B*d + 3*A*e) - 3*b*c*d*(B*d + 4*A*e))*Log[x] - (2*(c*d - b*e)^2*(3*b*B*c^2*d^2 - 6*A*c^3*d^2 +
2*b^2*B*c*d*e + b^3*B*e^2)*Log[b + c*x])/c^3)/b^5

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Maple [A]
time = 0.63, size = 404, normalized size = 1.72

method	result	size
default	\(-\frac {A \,b^{4} e^{4} c -6 A \,b^{2} c^{3} d^{2} e^{2}+8 A \,c^{4} d^{3} e b -3 A \,c^{5} d^{4}-2 B \,e^{4} b^{5}+4 B \,b^{4} d \,e^{3} c -4 B \,d^{3} e \,b^{2} c^{3}+2 B \,c^{4} d^{4} b}{c^{3} b^{4} \left (c x +b \right )}+\frac {\left (-6 A \,b^{2} c^{3} d^{2} e^{2}+12 A \,c^{4} d^{3} e b -6 A \,c^{5} d^{4}+B \,e^{4} b^{5}-4 B \,d^{3} e \,b^{2} c^{3}+3 B \,c^{4} d^{4} b \right ) \ln \left (c x +b \right )}{b^{5} c^{3}}-\frac {-A \,b^{4} e^{4} c +4 A \,b^{3} c^{2} d \,e^{3}-6 A \,b^{2} c^{3} d^{2} e^{2}+4 A \,c^{4} d^{3} e b -A \,c^{5} d^{4}+B \,e^{4} b^{5}-4 B \,b^{4} d \,e^{3} c +6 B \,b^{3} c^{2} d^{2} e^{2}-4 B \,d^{3} e \,b^{2} c^{3}+B \,c^{4} d^{4} b}{2 b^{3} c^{3} \left (c x +b \right )^{2}}-\frac {A \,d^{4}}{2 b^{3} x^{2}}+\frac {d^{2} \left (6 A \,b^{2} e^{2}-12 A b c d e +6 A \,c^{2} d^{2}+4 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {d^{3} \left (4 A b e -3 A c d +B b d \right )}{b^{4} x}\)	\(404\)
norman	\(\frac {-\frac {A \,d^{4}}{2 b}-\frac {d^{3} \left (4 A b e -2 A c d +B b d \right ) x}{b^{2}}-\frac {\left (A \,b^{4} e^{4} c -6 A \,b^{2} c^{3} d^{2} e^{2}+12 A \,c^{4} d^{3} e b -6 A \,c^{5} d^{4}-2 B \,e^{4} b^{5}+4 B \,b^{4} d \,e^{3} c -4 B \,d^{3} e \,b^{2} c^{3}+3 B \,c^{4} d^{4} b \right ) x^{3}}{b^{4} c^{2}}-\frac {\left (A \,b^{4} e^{4} c +4 A \,b^{3} c^{2} d \,e^{3}-18 A \,b^{2} c^{3} d^{2} e^{2}+36 A \,c^{4} d^{3} e b -18 A \,c^{5} d^{4}-3 B \,e^{4} b^{5}+4 B \,b^{4} d \,e^{3} c +6 B \,b^{3} c^{2} d^{2} e^{2}-12 B \,d^{3} e \,b^{2} c^{3}+9 B \,c^{4} d^{4} b \right ) x^{2}}{2 b^{3} c^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {d^{2} \left (6 A \,b^{2} e^{2}-12 A b c d e +6 A \,c^{2} d^{2}+4 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {\left (6 A \,b^{2} c^{3} d^{2} e^{2}-12 A \,c^{4} d^{3} e b +6 A \,c^{5} d^{4}-B \,e^{4} b^{5}+4 B \,d^{3} e \,b^{2} c^{3}-3 B \,c^{4} d^{4} b \right ) \ln \left (c x +b \right )}{b^{5} c^{3}}\)	\(406\)
risch	\(\frac {-\frac {A \,d^{4}}{2 b}-\frac {d^{3} \left (4 A b e -2 A c d +B b d \right ) x}{b^{2}}-\frac {\left (A \,b^{4} e^{4} c -6 A \,b^{2} c^{3} d^{2} e^{2}+12 A \,c^{4} d^{3} e b -6 A \,c^{5} d^{4}-2 B \,e^{4} b^{5}+4 B \,b^{4} d \,e^{3} c -4 B \,d^{3} e \,b^{2} c^{3}+3 B \,c^{4} d^{4} b \right ) x^{3}}{b^{4} c^{2}}-\frac {\left (A \,b^{4} e^{4} c +4 A \,b^{3} c^{2} d \,e^{3}-18 A \,b^{2} c^{3} d^{2} e^{2}+36 A \,c^{4} d^{3} e b -18 A \,c^{5} d^{4}-3 B \,e^{4} b^{5}+4 B \,b^{4} d \,e^{3} c +6 B \,b^{3} c^{2} d^{2} e^{2}-12 B \,d^{3} e \,b^{2} c^{3}+9 B \,c^{4} d^{4} b \right ) x^{2}}{2 b^{3} c^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {6 d^{2} \ln \left (-x \right ) A \,e^{2}}{b^{3}}-\frac {12 d^{3} \ln \left (-x \right ) A c e}{b^{4}}+\frac {6 d^{4} \ln \left (-x \right ) A \,c^{2}}{b^{5}}+\frac {4 d^{3} \ln \left (-x \right ) B e}{b^{3}}-\frac {3 d^{4} \ln \left (-x \right ) B c}{b^{4}}-\frac {6 \ln \left (c x +b \right ) A \,d^{2} e^{2}}{b^{3}}+\frac {12 c \ln \left (c x +b \right ) A \,d^{3} e}{b^{4}}-\frac {6 c^{2} \ln \left (c x +b \right ) A \,d^{4}}{b^{5}}+\frac {\ln \left (c x +b \right ) B \,e^{4}}{c^{3}}-\frac {4 \ln \left (c x +b \right ) B \,d^{3} e}{b^{3}}+\frac {3 c \ln \left (c x +b \right ) B \,d^{4}}{b^{4}}\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 438, normalized size = 1.86 \begin {gather*} -\frac {A b^{3} c^{3} d^{4} - 2 \, {\left (6 \, A b^{2} c^{4} d^{2} e^{2} - 4 \, B b^{4} c^{2} d e^{3} + 2 \, B b^{5} c e^{4} - A b^{4} c^{2} e^{4} - 3 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \, {\left (B b^{2} c^{4} e - 3 \, A b c^{5} e\right )} d^{3}\right )} x^{3} - {\left (3 \, B b^{6} e^{4} - A b^{5} c e^{4} - 9 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} + 12 \, {\left (B b^{3} c^{3} e - 3 \, A b^{2} c^{4} e\right )} d^{3} - 6 \, {\left (B b^{4} c^{2} e^{2} - 3 \, A b^{3} c^{3} e^{2}\right )} d^{2} - 4 \, {\left (B b^{5} c e^{3} + A b^{4} c^{2} e^{3}\right )} d\right )} x^{2} + 2 \, {\left (4 \, A b^{3} c^{3} d^{3} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{4}\right )} x}{2 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac {{\left (6 \, A b^{2} d^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} e - 3 \, A b c e\right )} d^{3}\right )} \log \left (x\right )}{b^{5}} - \frac {{\left (6 \, A b^{2} c^{3} d^{2} e^{2} - B b^{5} e^{4} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \, {\left (B b^{2} c^{3} e - 3 \, A b c^{4} e\right )} d^{3}\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(A*b^3*c^3*d^4 - 2*(6*A*b^2*c^4*d^2*e^2 - 4*B*b^4*c^2*d*e^3 + 2*B*b^5*c*e^4 - A*b^4*c^2*e^4 - 3*(B*b*c^5
- 2*A*c^6)*d^4 + 4*(B*b^2*c^4*e - 3*A*b*c^5*e)*d^3)*x^3 - (3*B*b^6*e^4 - A*b^5*c*e^4 - 9*(B*b^2*c^4 - 2*A*b*c^
5)*d^4 + 12*(B*b^3*c^3*e - 3*A*b^2*c^4*e)*d^3 - 6*(B*b^4*c^2*e^2 - 3*A*b^3*c^3*e^2)*d^2 - 4*(B*b^5*c*e^3 + A*b
^4*c^2*e^3)*d)*x^2 + 2*(4*A*b^3*c^3*d^3*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^4)*x)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b
^6*c^3*x^2) + (6*A*b^2*d^2*e^2 - 3*(B*b*c - 2*A*c^2)*d^4 + 4*(B*b^2*e - 3*A*b*c*e)*d^3)*log(x)/b^5 - (6*A*b^2*
c^3*d^2*e^2 - B*b^5*e^4 - 3*(B*b*c^4 - 2*A*c^5)*d^4 + 4*(B*b^2*c^3*e - 3*A*b*c^4*e)*d^3)*log(c*x + b)/(b^5*c^3
)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (240) = 480\).
time = 3.87, size = 764, normalized size = 3.25 \begin {gather*} -\frac {A b^{4} c^{3} d^{4} + 6 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} x^{3} + 9 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4} x^{2} + 2 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} d^{4} x - {\left (2 \, {\left (2 \, B b^{6} c - A b^{5} c^{2}\right )} x^{3} + {\left (3 \, B b^{7} - A b^{6} c\right )} x^{2}\right )} e^{4} + 4 \, {\left (2 \, B b^{5} c^{2} d x^{3} + {\left (B b^{6} c + A b^{5} c^{2}\right )} d x^{2}\right )} e^{3} - 6 \, {\left (2 \, A b^{3} c^{4} d^{2} x^{3} - {\left (B b^{5} c^{2} - 3 \, A b^{4} c^{3}\right )} d^{2} x^{2}\right )} e^{2} + 4 \, {\left (2 \, A b^{4} c^{3} d^{3} x - 2 \, {\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{3} x^{3} - 3 \, {\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} x^{2}\right )} e - 2 \, {\left (3 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} x^{4} + 6 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} x^{3} + 3 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4} x^{2} + {\left (B b^{5} c^{2} x^{4} + 2 \, B b^{6} c x^{3} + B b^{7} x^{2}\right )} e^{4} - 6 \, {\left (A b^{2} c^{5} d^{2} x^{4} + 2 \, A b^{3} c^{4} d^{2} x^{3} + A b^{4} c^{3} d^{2} x^{2}\right )} e^{2} - 4 \, {\left ({\left (B b^{2} c^{5} - 3 \, A b c^{6}\right )} d^{3} x^{4} + 2 \, {\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{3} x^{3} + {\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} x^{2}\right )} e\right )} \log \left (c x + b\right ) + 2 \, {\left (3 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} x^{4} + 6 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} x^{3} + 3 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4} x^{2} - 6 \, {\left (A b^{2} c^{5} d^{2} x^{4} + 2 \, A b^{3} c^{4} d^{2} x^{3} + A b^{4} c^{3} d^{2} x^{2}\right )} e^{2} - 4 \, {\left ({\left (B b^{2} c^{5} - 3 \, A b c^{6}\right )} d^{3} x^{4} + 2 \, {\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{3} x^{3} + {\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} x^{2}\right )} e\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{5} x^{4} + 2 \, b^{6} c^{4} x^{3} + b^{7} c^{3} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(A*b^4*c^3*d^4 + 6*(B*b^2*c^5 - 2*A*b*c^6)*d^4*x^3 + 9*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4*x^2 + 2*(B*b^4*c^3 -
2*A*b^3*c^4)*d^4*x - (2*(2*B*b^6*c - A*b^5*c^2)*x^3 + (3*B*b^7 - A*b^6*c)*x^2)*e^4 + 4*(2*B*b^5*c^2*d*x^3 + (
B*b^6*c + A*b^5*c^2)*d*x^2)*e^3 - 6*(2*A*b^3*c^4*d^2*x^3 - (B*b^5*c^2 - 3*A*b^4*c^3)*d^2*x^2)*e^2 + 4*(2*A*b^4
*c^3*d^3*x - 2*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*x^3 - 3*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*x^2)*e - 2*(3*(B*b*c^6 - 2*
A*c^7)*d^4*x^4 + 6*(B*b^2*c^5 - 2*A*b*c^6)*d^4*x^3 + 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4*x^2 + (B*b^5*c^2*x^4 + 2*
B*b^6*c*x^3 + B*b^7*x^2)*e^4 - 6*(A*b^2*c^5*d^2*x^4 + 2*A*b^3*c^4*d^2*x^3 + A*b^4*c^3*d^2*x^2)*e^2 - 4*((B*b^2
*c^5 - 3*A*b*c^6)*d^3*x^4 + 2*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*x^3 + (B*b^4*c^3 - 3*A*b^3*c^4)*d^3*x^2)*e)*log(c*
x + b) + 2*(3*(B*b*c^6 - 2*A*c^7)*d^4*x^4 + 6*(B*b^2*c^5 - 2*A*b*c^6)*d^4*x^3 + 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^
4*x^2 - 6*(A*b^2*c^5*d^2*x^4 + 2*A*b^3*c^4*d^2*x^3 + A*b^4*c^3*d^2*x^2)*e^2 - 4*((B*b^2*c^5 - 3*A*b*c^6)*d^3*x
^4 + 2*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*x^3 + (B*b^4*c^3 - 3*A*b^3*c^4)*d^3*x^2)*e)*log(x))/(b^5*c^5*x^4 + 2*b^6*
c^4*x^3 + b^7*c^3*x^2)

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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)*(e*x+d)**4/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.86, size = 428, normalized size = 1.82 \begin {gather*} -\frac {{\left (3 \, B b c d^{4} - 6 \, A c^{2} d^{4} - 4 \, B b^{2} d^{3} e + 12 \, A b c d^{3} e - 6 \, A b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (3 \, B b c^{4} d^{4} - 6 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 12 \, A b c^{4} d^{3} e - 6 \, A b^{2} c^{3} d^{2} e^{2} + B b^{5} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac {A b^{3} c^{3} d^{4} + 2 \, {\left (3 \, B b c^{5} d^{4} - 6 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 12 \, A b c^{5} d^{3} e - 6 \, A b^{2} c^{4} d^{2} e^{2} + 4 \, B b^{4} c^{2} d e^{3} - 2 \, B b^{5} c e^{4} + A b^{4} c^{2} e^{4}\right )} x^{3} + {\left (9 \, B b^{2} c^{4} d^{4} - 18 \, A b c^{5} d^{4} - 12 \, B b^{3} c^{3} d^{3} e + 36 \, A b^{2} c^{4} d^{3} e + 6 \, B b^{4} c^{2} d^{2} e^{2} - 18 \, A b^{3} c^{3} d^{2} e^{2} + 4 \, B b^{5} c d e^{3} + 4 \, A b^{4} c^{2} d e^{3} - 3 \, B b^{6} e^{4} + A b^{5} c e^{4}\right )} x^{2} + 2 \, {\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-(3*B*b*c*d^4 - 6*A*c^2*d^4 - 4*B*b^2*d^3*e + 12*A*b*c*d^3*e - 6*A*b^2*d^2*e^2)*log(abs(x))/b^5 + (3*B*b*c^4*d
^4 - 6*A*c^5*d^4 - 4*B*b^2*c^3*d^3*e + 12*A*b*c^4*d^3*e - 6*A*b^2*c^3*d^2*e^2 + B*b^5*e^4)*log(abs(c*x + b))/(
b^5*c^3) - 1/2*(A*b^3*c^3*d^4 + 2*(3*B*b*c^5*d^4 - 6*A*c^6*d^4 - 4*B*b^2*c^4*d^3*e + 12*A*b*c^5*d^3*e - 6*A*b^
2*c^4*d^2*e^2 + 4*B*b^4*c^2*d*e^3 - 2*B*b^5*c*e^4 + A*b^4*c^2*e^4)*x^3 + (9*B*b^2*c^4*d^4 - 18*A*b*c^5*d^4 - 1
2*B*b^3*c^3*d^3*e + 36*A*b^2*c^4*d^3*e + 6*B*b^4*c^2*d^2*e^2 - 18*A*b^3*c^3*d^2*e^2 + 4*B*b^5*c*d*e^3 + 4*A*b^
4*c^2*d*e^3 - 3*B*b^6*e^4 + A*b^5*c*e^4)*x^2 + 2*(B*b^3*c^3*d^4 - 2*A*b^2*c^4*d^4 + 4*A*b^3*c^3*d^3*e)*x)/((c*
x + b)^2*b^4*c^3*x^2)

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Mupad [B]
time = 1.80, size = 403, normalized size = 1.71 \begin {gather*} \frac {\ln \left (x\right )\,\left (b^2\,\left (4\,B\,d^3\,e+6\,A\,d^2\,e^2\right )-b\,\left (3\,B\,c\,d^4+12\,A\,c\,e\,d^3\right )+6\,A\,c^2\,d^4\right )}{b^5}-\frac {\frac {A\,d^4}{2\,b}+\frac {x^2\,\left (-3\,B\,b^5\,e^4+4\,B\,b^4\,c\,d\,e^3+A\,b^4\,c\,e^4+6\,B\,b^3\,c^2\,d^2\,e^2+4\,A\,b^3\,c^2\,d\,e^3-12\,B\,b^2\,c^3\,d^3\,e-18\,A\,b^2\,c^3\,d^2\,e^2+9\,B\,b\,c^4\,d^4+36\,A\,b\,c^4\,d^3\,e-18\,A\,c^5\,d^4\right )}{2\,b^3\,c^3}-\frac {x^3\,\left (2\,B\,b^5\,e^4-4\,B\,b^4\,c\,d\,e^3-A\,b^4\,c\,e^4+4\,B\,b^2\,c^3\,d^3\,e+6\,A\,b^2\,c^3\,d^2\,e^2-3\,B\,b\,c^4\,d^4-12\,A\,b\,c^4\,d^3\,e+6\,A\,c^5\,d^4\right )}{b^4\,c^2}+\frac {d^3\,x\,\left (4\,A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^2\,\left (B\,b^3\,e^2+2\,B\,b^2\,c\,d\,e+3\,B\,b\,c^2\,d^2-6\,A\,c^3\,d^2\right )}{b^5\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^3,x)

[Out]

(log(x)*(b^2*(6*A*d^2*e^2 + 4*B*d^3*e) - b*(3*B*c*d^4 + 12*A*c*d^3*e) + 6*A*c^2*d^4))/b^5 - ((A*d^4)/(2*b) + (
x^2*(A*b^4*c*e^4 - 3*B*b^5*e^4 - 18*A*c^5*d^4 + 9*B*b*c^4*d^4 + 4*A*b^3*c^2*d*e^3 - 12*B*b^2*c^3*d^3*e - 18*A*
b^2*c^3*d^2*e^2 + 6*B*b^3*c^2*d^2*e^2 + 36*A*b*c^4*d^3*e + 4*B*b^4*c*d*e^3))/(2*b^3*c^3) - (x^3*(6*A*c^5*d^4 +
2*B*b^5*e^4 - A*b^4*c*e^4 - 3*B*b*c^4*d^4 + 4*B*b^2*c^3*d^3*e + 6*A*b^2*c^3*d^2*e^2 - 12*A*b*c^4*d^3*e - 4*B*
b^4*c*d*e^3))/(b^4*c^2) + (d^3*x*(4*A*b*e - 2*A*c*d + B*b*d))/b^2)/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) + (log(b +
c*x)*(b*e - c*d)^2*(B*b^3*e^2 - 6*A*c^3*d^2 + 3*B*b*c^2*d^2 + 2*B*b^2*c*d*e))/(b^5*c^3)

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