3.3.83 \(\int \frac {1}{(d+e x) (b x+c x^2)^3} \, dx\) [283]

Optimal. Leaf size=193 \[ -\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}+\frac {c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]

[Out]

-1/2/b^3/d/x^2+(b*e+3*c*d)/b^4/d^2/x+1/2*c^3/b^3/(-b*e+c*d)/(c*x+b)^2+c^3*(-4*b*e+3*c*d)/b^4/(-b*e+c*d)^2/(c*x
+b)+(b^2*e^2+3*b*c*d*e+6*c^2*d^2)*ln(x)/b^5/d^3-c^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^2)*ln(c*x+b)/b^5/(-b*e+c*d)
^3+e^5*ln(e*x+d)/d^3/(-b*e+c*d)^3

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Rubi [A]
time = 0.16, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \begin {gather*} \frac {c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac {b e+3 c d}{b^4 d^2 x}+\frac {c^3}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {1}{2 b^3 d x^2}+\frac {\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac {c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)^3),x]

[Out]

-1/2*1/(b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) + c^3/(2*b^3*(c*d - b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/(
b^4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) - (c^3*(6*c^2*d^2 - 15*b*c
*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)

Rule 712

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {1}{b^3 d x^3}+\frac {-3 c d-b e}{b^4 d^2 x^2}+\frac {6 c^2 d^2+3 b c d e+b^2 e^2}{b^5 d^3 x}+\frac {c^4}{b^3 (-c d+b e) (b+c x)^3}+\frac {c^4 (-3 c d+4 b e)}{b^4 (-c d+b e)^2 (b+c x)^2}+\frac {c^4 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 (-c d+b e)^3 (b+c x)}+\frac {e^6}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}+\frac {c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 192, normalized size = 0.99 \begin {gather*} -\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}-\frac {c^3}{2 b^3 (-c d+b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}+\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (-c d+b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(b*x + c*x^2)^3),x]

[Out]

-1/2*1/(b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) - c^3/(2*b^3*(-(c*d) + b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e)
)/(b^4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) + (c^3*(6*c^2*d^2 - 15*
b*c*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(-(c*d) + b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)

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Maple [A]
time = 0.55, size = 193, normalized size = 1.00

method	result	size
default	\(-\frac {c^{3}}{2 \left (b e -c d \right ) b^{3} \left (c x +b \right )^{2}}-\frac {c^{3} \left (4 b e -3 c d \right )}{\left (b e -c d \right )^{2} b^{4} \left (c x +b \right )}+\frac {c^{3} \left (10 b^{2} e^{2}-15 b c d e +6 d^{2} c^{2}\right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{5}}-\frac {1}{2 b^{3} d \,x^{2}}-\frac {-b e -3 c d}{b^{4} d^{2} x}+\frac {\left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}-\frac {e^{5} \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}\)	\(193\)
norman	\(\frac {\frac {\left (b e +2 c d \right ) x}{b^{2} d^{2}}+\frac {\left (-3 b^{3} c \,e^{3}-2 b^{2} c^{2} d \,e^{2}+18 b \,c^{3} d^{2} e -12 c^{4} d^{3}\right ) c \,x^{3}}{d^{2} b^{4} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}-\frac {1}{2 b d}+\frac {c^{2} \left (-4 b^{3} c \,e^{3}-3 b^{2} c^{2} d \,e^{2}+27 b \,c^{3} d^{2} e -18 c^{4} d^{3}\right ) x^{4}}{2 d^{2} b^{5} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}+\frac {c^{3} \left (10 b^{2} e^{2}-15 b c d e +6 d^{2} c^{2}\right ) \ln \left (c x +b \right )}{b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {e^{5} \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\)	\(346\)
risch	\(\frac {\frac {c^{2} \left (b^{3} e^{3}+b^{2} d \,e^{2} c -9 b \,c^{2} d^{2} e +6 c^{3} d^{3}\right ) x^{3}}{b^{4} d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}+\frac {c \left (4 b^{3} e^{3}+3 b^{2} d \,e^{2} c -27 b \,c^{2} d^{2} e +18 c^{3} d^{3}\right ) x^{2}}{2 b^{3} d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}+\frac {\left (b e +2 c d \right ) x}{b^{2} d^{2}}-\frac {1}{2 b d}}{x^{2} \left (c x +b \right )^{2}}+\frac {\ln \left (-x \right ) e^{2}}{d^{3} b^{3}}+\frac {3 \ln \left (-x \right ) c e}{d^{2} b^{4}}+\frac {6 \ln \left (-x \right ) c^{2}}{d \,b^{5}}-\frac {e^{5} \ln \left (-e x -d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {10 c^{3} \ln \left (c x +b \right ) e^{2}}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {15 c^{4} \ln \left (c x +b \right ) d e}{b^{4} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {6 c^{5} \ln \left (c x +b \right ) d^{2}}{b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*c^3/(b*e-c*d)/b^3/(c*x+b)^2-c^3*(4*b*e-3*c*d)/(b*e-c*d)^2/b^4/(c*x+b)+c^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^
2)/(b*e-c*d)^3/b^5*ln(c*x+b)-1/2/b^3/d/x^2-(-b*e-3*c*d)/b^4/d^2/x+(b^2*e^2+3*b*c*d*e+6*c^2*d^2)*ln(x)/b^5/d^3-
e^5/d^3/(b*e-c*d)^3*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (195) = 390\).
time = 0.31, size = 435, normalized size = 2.25 \begin {gather*} -\frac {{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} + \frac {e^{5} \log \left (x e + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \, {\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} - {\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-(6*c^5*d^2 - 15*b*c^4*d*e + 10*b^2*c^3*e^2)*log(c*x + b)/(b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8
*e^3) + e^5*log(x*e + d)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3) - 1/2*(b^3*c^2*d^3 - 2*b^4*
c*d^2*e + b^5*d*e^2 - 2*(6*c^5*d^3 - 9*b*c^4*d^2*e + b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^3 - (18*b*c^4*d^3 - 27*b^2
*c^3*d^2*e + 3*b^3*c^2*d*e^2 + 4*b^4*c*e^3)*x^2 - 2*(2*b^2*c^3*d^3 - 3*b^3*c^2*d^2*e + b^5*e^3)*x)/((b^4*c^4*d
^4 - 2*b^5*c^3*d^3*e + b^6*c^2*d^2*e^2)*x^4 + 2*(b^5*c^3*d^4 - 2*b^6*c^2*d^3*e + b^7*c*d^2*e^2)*x^3 + (b^6*c^2
*d^4 - 2*b^7*c*d^3*e + b^8*d^2*e^2)*x^2) + (6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (195) = 390\).
time = 31.91, size = 731, normalized size = 3.79 \begin {gather*} \frac {12 \, b c^{6} d^{5} x^{3} + 18 \, b^{2} c^{5} d^{5} x^{2} + 4 \, b^{3} c^{4} d^{5} x - b^{4} c^{3} d^{5} + 2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )} e^{5} \log \left (x e + d\right ) - 2 \, {\left (b^{5} c^{2} d x^{3} + 2 \, b^{6} c d x^{2} + b^{7} d x\right )} e^{4} + {\left (b^{5} c^{2} d^{2} x^{2} + 2 \, b^{6} c d^{2} x + b^{7} d^{2}\right )} e^{3} + {\left (20 \, b^{3} c^{4} d^{3} x^{3} + 30 \, b^{4} c^{3} d^{3} x^{2} + 6 \, b^{5} c^{2} d^{3} x - 3 \, b^{6} c d^{3}\right )} e^{2} - {\left (30 \, b^{2} c^{5} d^{4} x^{3} + 45 \, b^{3} c^{4} d^{4} x^{2} + 10 \, b^{4} c^{3} d^{4} x - 3 \, b^{5} c^{2} d^{4}\right )} e - 2 \, {\left (6 \, c^{7} d^{5} x^{4} + 12 \, b c^{6} d^{5} x^{3} + 6 \, b^{2} c^{5} d^{5} x^{2} + 10 \, {\left (b^{2} c^{5} d^{3} x^{4} + 2 \, b^{3} c^{4} d^{3} x^{3} + b^{4} c^{3} d^{3} x^{2}\right )} e^{2} - 15 \, {\left (b c^{6} d^{4} x^{4} + 2 \, b^{2} c^{5} d^{4} x^{3} + b^{3} c^{4} d^{4} x^{2}\right )} e\right )} \log \left (c x + b\right ) + 2 \, {\left (6 \, c^{7} d^{5} x^{4} + 12 \, b c^{6} d^{5} x^{3} + 6 \, b^{2} c^{5} d^{5} x^{2} - {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )} e^{5} + 10 \, {\left (b^{2} c^{5} d^{3} x^{4} + 2 \, b^{3} c^{4} d^{3} x^{3} + b^{4} c^{3} d^{3} x^{2}\right )} e^{2} - 15 \, {\left (b c^{6} d^{4} x^{4} + 2 \, b^{2} c^{5} d^{4} x^{3} + b^{3} c^{4} d^{4} x^{2}\right )} e\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{5} d^{6} x^{4} + 2 \, b^{6} c^{4} d^{6} x^{3} + b^{7} c^{3} d^{6} x^{2} - {\left (b^{8} c^{2} d^{3} x^{4} + 2 \, b^{9} c d^{3} x^{3} + b^{10} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (b^{7} c^{3} d^{4} x^{4} + 2 \, b^{8} c^{2} d^{4} x^{3} + b^{9} c d^{4} x^{2}\right )} e^{2} - 3 \, {\left (b^{6} c^{4} d^{5} x^{4} + 2 \, b^{7} c^{3} d^{5} x^{3} + b^{8} c^{2} d^{5} x^{2}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/2*(12*b*c^6*d^5*x^3 + 18*b^2*c^5*d^5*x^2 + 4*b^3*c^4*d^5*x - b^4*c^3*d^5 + 2*(b^5*c^2*x^4 + 2*b^6*c*x^3 + b^
7*x^2)*e^5*log(x*e + d) - 2*(b^5*c^2*d*x^3 + 2*b^6*c*d*x^2 + b^7*d*x)*e^4 + (b^5*c^2*d^2*x^2 + 2*b^6*c*d^2*x +
b^7*d^2)*e^3 + (20*b^3*c^4*d^3*x^3 + 30*b^4*c^3*d^3*x^2 + 6*b^5*c^2*d^3*x - 3*b^6*c*d^3)*e^2 - (30*b^2*c^5*d^
4*x^3 + 45*b^3*c^4*d^4*x^2 + 10*b^4*c^3*d^4*x - 3*b^5*c^2*d^4)*e - 2*(6*c^7*d^5*x^4 + 12*b*c^6*d^5*x^3 + 6*b^2
*c^5*d^5*x^2 + 10*(b^2*c^5*d^3*x^4 + 2*b^3*c^4*d^3*x^3 + b^4*c^3*d^3*x^2)*e^2 - 15*(b*c^6*d^4*x^4 + 2*b^2*c^5*
d^4*x^3 + b^3*c^4*d^4*x^2)*e)*log(c*x + b) + 2*(6*c^7*d^5*x^4 + 12*b*c^6*d^5*x^3 + 6*b^2*c^5*d^5*x^2 - (b^5*c^
2*x^4 + 2*b^6*c*x^3 + b^7*x^2)*e^5 + 10*(b^2*c^5*d^3*x^4 + 2*b^3*c^4*d^3*x^3 + b^4*c^3*d^3*x^2)*e^2 - 15*(b*c^
6*d^4*x^4 + 2*b^2*c^5*d^4*x^3 + b^3*c^4*d^4*x^2)*e)*log(x))/(b^5*c^5*d^6*x^4 + 2*b^6*c^4*d^6*x^3 + b^7*c^3*d^6
*x^2 - (b^8*c^2*d^3*x^4 + 2*b^9*c*d^3*x^3 + b^10*d^3*x^2)*e^3 + 3*(b^7*c^3*d^4*x^4 + 2*b^8*c^2*d^4*x^3 + b^9*c
*d^4*x^2)*e^2 - 3*(b^6*c^4*d^5*x^4 + 2*b^7*c^3*d^5*x^3 + b^8*c^2*d^5*x^2)*e)

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (195) = 390\).
time = 2.72, size = 414, normalized size = 2.15 \begin {gather*} -\frac {{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac {e^{6} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-(6*c^6*d^2 - 15*b*c^5*d*e + 10*b^2*c^4*e^2)*log(abs(c*x + b))/(b^5*c^4*d^3 - 3*b^6*c^3*d^2*e + 3*b^7*c^2*d*e^
2 - b^8*c*e^3) + e^6*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*e^4) + (6*c^2*
d^2 + 3*b*c*d*e + b^2*e^2)*log(abs(x))/(b^5*d^3) - 1/2*(b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*
d^2*e^3 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - b^4*c^2*d*e^4)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4
*d^4*e + 30*b^3*c^3*d^3*e^2 + b^4*c^2*d^2*e^3 - 4*b^5*c*d*e^4)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e + 3*b^
4*c^2*d^3*e^2 + b^5*c*d^2*e^3 - b^6*d*e^4)*x)/((c*d - b*e)^3*(c*x + b)^2*b^4*d^3*x^2)

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Mupad [B]
time = 0.83, size = 331, normalized size = 1.72 \begin {gather*} \frac {\frac {x\,\left (b\,e+2\,c\,d\right )}{b^2\,d^2}-\frac {1}{2\,b\,d}+\frac {x^2\,\left (4\,b^3\,c\,e^3+3\,b^2\,c^2\,d\,e^2-27\,b\,c^3\,d^2\,e+18\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^3\,\left (b^3\,c^2\,e^3+b^2\,c^3\,d\,e^2-9\,b\,c^4\,d^2\,e+6\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,\left (10\,b^2\,c^3\,e^2-15\,b\,c^4\,d\,e+6\,c^5\,d^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}-\frac {e^5\,\ln \left (d+e\,x\right )}{d^3\,{\left (b\,e-c\,d\right )}^3}+\frac {\ln \left (x\right )\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^3*(d + e*x)),x)

[Out]

((x*(b*e + 2*c*d))/(b^2*d^2) - 1/(2*b*d) + (x^2*(18*c^4*d^3 + 4*b^3*c*e^3 + 3*b^2*c^2*d*e^2 - 27*b*c^3*d^2*e))
/(2*b^3*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)) + (x^3*(6*c^5*d^3 + b^3*c^2*e^3 + b^2*c^3*d*e^2 - 9*b*c^4*d^2*e))
/(b^4*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)))/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) + (log(b + c*x)*(6*c^5*d^2 + 10*b^
2*c^3*e^2 - 15*b*c^4*d*e))/(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2) - (e^5*log(d + e*x))/(d^3
*(b*e - c*d)^3) + (log(x)*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(b^5*d^3)

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