3.275 \(\int \frac{(d+e x)^7}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=203 \[ \frac{3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac{(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}+\frac{d^6 (3 c d-7 b e)}{b^4 x}-\frac{d^7}{2 b^3 x^2}+\frac{e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{2 c^3} \]

[Out]

-d^7/(2*b^3*x^2) + (d^6*(3*c*d - 7*b*e))/(b^4*x) + (e^6*(7*c*d - 3*b*e)*x)/c^4 + (e^7*x^2)/(2*c^3) + (c*d - b*
e)^7/(2*b^3*c^5*(b + c*x)^2) + ((c*d - b*e)^6*(3*c*d + 4*b*e))/(b^4*c^5*(b + c*x)) + (3*d^5*(2*c^2*d^2 - 7*b*c
*d*e + 7*b^2*e^2)*Log[x])/b^5 - (3*(c*d - b*e)^5*(2*c^2*d^2 + 3*b*c*d*e + 2*b^2*e^2)*Log[b + c*x])/(b^5*c^5)

________________________________________________________________________________________

Rubi [A]  time = 0.268947, antiderivative size = 203, normalized size of antiderivative = 1., 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 = {698} \[ \frac{3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac{(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}+\frac{d^6 (3 c d-7 b e)}{b^4 x}-\frac{d^7}{2 b^3 x^2}+\frac{e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^7/(b*x + c*x^2)^3,x]

[Out]

-d^7/(2*b^3*x^2) + (d^6*(3*c*d - 7*b*e))/(b^4*x) + (e^6*(7*c*d - 3*b*e)*x)/c^4 + (e^7*x^2)/(2*c^3) + (c*d - b*
e)^7/(2*b^3*c^5*(b + c*x)^2) + ((c*d - b*e)^6*(3*c*d + 4*b*e))/(b^4*c^5*(b + c*x)) + (3*d^5*(2*c^2*d^2 - 7*b*c
*d*e + 7*b^2*e^2)*Log[x])/b^5 - (3*(c*d - b*e)^5*(2*c^2*d^2 + 3*b*c*d*e + 2*b^2*e^2)*Log[b + c*x])/(b^5*c^5)

Rule 698

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

Mathematica [A]  time = 0.123366, size = 202, normalized size = 1. \[ \frac{1}{2} \left (\frac{6 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}+\frac{6 (b e-c d)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac{2 (c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{(c d-b e)^7}{b^3 c^5 (b+c x)^2}+\frac{2 d^6 (3 c d-7 b e)}{b^4 x}-\frac{d^7}{b^3 x^2}+\frac{2 e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{c^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^7/(b*x + c*x^2)^3,x]

[Out]

(-(d^7/(b^3*x^2)) + (2*d^6*(3*c*d - 7*b*e))/(b^4*x) + (2*e^6*(7*c*d - 3*b*e)*x)/c^4 + (e^7*x^2)/c^3 + (c*d - b
*e)^7/(b^3*c^5*(b + c*x)^2) + (2*(c*d - b*e)^6*(3*c*d + 4*b*e))/(b^4*c^5*(b + c*x)) + (6*d^5*(2*c^2*d^2 - 7*b*
c*d*e + 7*b^2*e^2)*Log[x])/b^5 + (6*(-(c*d) + b*e)^5*(2*c^2*d^2 + 3*b*c*d*e + 2*b^2*e^2)*Log[b + c*x])/(b^5*c^
5))/2

________________________________________________________________________________________

Maple [B]  time = 0.066, size = 481, normalized size = 2.4 \begin{align*} -{\frac{{d}^{7}}{2\,{b}^{3}{x}^{2}}}+{\frac{{e}^{7}{x}^{2}}{2\,{c}^{3}}}-3\,{\frac{{e}^{7}xb}{{c}^{4}}}+7\,{\frac{d{e}^{6}x}{{c}^{3}}}-7\,{\frac{{d}^{6}e}{{b}^{3}x}}+3\,{\frac{{d}^{7}c}{{b}^{4}x}}+21\,{\frac{{d}^{5}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{7}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}+6\,{\frac{{b}^{2}\ln \left ( cx+b \right ){e}^{7}}{{c}^{5}}}+21\,{\frac{\ln \left ( cx+b \right ){d}^{2}{e}^{5}}{{c}^{3}}}-21\,{\frac{\ln \left ( cx+b \right ){d}^{5}{e}^{2}}{{b}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ){d}^{7}}{{b}^{5}}}+4\,{\frac{{b}^{3}{e}^{7}}{{c}^{5} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{7}}{{b}^{4} \left ( cx+b \right ) }}-{\frac{{b}^{4}{e}^{7}}{2\,{c}^{5} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{7}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-35\,{\frac{{d}^{3}{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+21\,{\frac{{d}^{5}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-{\frac{35\,{d}^{4}{e}^{3}}{2\,c \left ( cx+b \right ) ^{2}}}+{\frac{21\,{d}^{5}{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{21\,{b}^{2}{d}^{2}{e}^{5}}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{35\,b{d}^{3}{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-{\frac{7\,c{d}^{6}e}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}-21\,{\frac{b\ln \left ( cx+b \right ) d{e}^{6}}{{c}^{4}}}+21\,{\frac{c\ln \left ( cx+b \right ){d}^{6}e}{{b}^{4}}}-21\,{\frac{{b}^{2}d{e}^{6}}{{c}^{4} \left ( cx+b \right ) }}+42\,{\frac{b{d}^{2}{e}^{5}}{{c}^{3} \left ( cx+b \right ) }}-21\,{\frac{{d}^{6}\ln \left ( x \right ) ce}{{b}^{4}}}-14\,{\frac{c{d}^{6}e}{{b}^{3} \left ( cx+b \right ) }}+{\frac{7\,{b}^{3}d{e}^{6}}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7/(c*x^2+b*x)^3,x)

[Out]

-1/2*d^7/b^3/x^2+1/2*e^7*x^2/c^3-3*e^7/c^4*x*b+7*e^6/c^3*d*x-7*d^6/b^3/x*e+3*d^7/b^4/x*c+21*d^5/b^3*ln(x)*e^2+
6*d^7/b^5*ln(x)*c^2+6/c^5*b^2*ln(c*x+b)*e^7+21/c^3*ln(c*x+b)*d^2*e^5-21/b^3*ln(c*x+b)*d^5*e^2-6*c^2/b^5*ln(c*x
+b)*d^7+4/c^5*b^3/(c*x+b)*e^7+3*c^2/b^4/(c*x+b)*d^7-1/2/c^5*b^4/(c*x+b)^2*e^7+1/2*c^2/b^3/(c*x+b)^2*d^7-35/c^2
/(c*x+b)*d^3*e^4+21/b^2/(c*x+b)*d^5*e^2-35/2/c/(c*x+b)^2*d^4*e^3+21/2/b/(c*x+b)^2*d^5*e^2-21/2/c^3*b^2/(c*x+b)
^2*d^2*e^5+35/2/c^2*b/(c*x+b)^2*d^3*e^4-7/2*c/b^2/(c*x+b)^2*d^6*e-21/c^4*b*ln(c*x+b)*d*e^6+21*c/b^4*ln(c*x+b)*
d^6*e-21/c^4*b^2/(c*x+b)*d*e^6+42/c^3*b/(c*x+b)*d^2*e^5-21*d^6/b^4*ln(x)*c*e-14*c/b^3/(c*x+b)*d^6*e+7/2/c^4*b^
3/(c*x+b)^2*d*e^6

________________________________________________________________________________________

Maxima [B]  time = 1.17236, size = 551, normalized size = 2.71 \begin{align*} -\frac{b^{3} c^{5} d^{7} - 2 \,{\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} -{\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \,{\left (b^{4} c^{7} x^{4} + 2 \, b^{5} c^{6} x^{3} + b^{6} c^{5} x^{2}\right )}} + \frac{c e^{7} x^{2} + 2 \,{\left (7 \, c d e^{6} - 3 \, b e^{7}\right )} x}{2 \, c^{4}} + \frac{3 \,{\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left (c x + b\right )}{b^{5} c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*c^5*d^7 - 2*(6*c^8*d^7 - 21*b*c^7*d^6*e + 21*b^2*c^6*d^5*e^2 - 35*b^4*c^4*d^3*e^4 + 42*b^5*c^3*d^2*e
^5 - 21*b^6*c^2*d*e^6 + 4*b^7*c*e^7)*x^3 - (18*b*c^7*d^7 - 63*b^2*c^6*d^6*e + 63*b^3*c^5*d^5*e^2 - 35*b^4*c^4*
d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 63*b^6*c^2*d^2*e^5 - 35*b^7*c*d*e^6 + 7*b^8*e^7)*x^2 - 2*(2*b^2*c^6*d^7 - 7*b^3
*c^5*d^6*e)*x)/(b^4*c^7*x^4 + 2*b^5*c^6*x^3 + b^6*c^5*x^2) + 1/2*(c*e^7*x^2 + 2*(7*c*d*e^6 - 3*b*e^7)*x)/c^4 +
3*(2*c^2*d^7 - 7*b*c*d^6*e + 7*b^2*d^5*e^2)*log(x)/b^5 - 3*(2*c^7*d^7 - 7*b*c^6*d^6*e + 7*b^2*c^5*d^5*e^2 - 7
*b^5*c^2*d^2*e^5 + 7*b^6*c*d*e^6 - 2*b^7*e^7)*log(c*x + b)/(b^5*c^5)

________________________________________________________________________________________

Fricas [B]  time = 2.12712, size = 1384, normalized size = 6.82 \begin{align*} \frac{b^{5} c^{4} e^{7} x^{6} - b^{4} c^{5} d^{7} + 2 \,{\left (7 \, b^{5} c^{4} d e^{6} - 2 \, b^{6} c^{3} e^{7}\right )} x^{5} +{\left (28 \, b^{6} c^{3} d e^{6} - 11 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \,{\left (6 \, b c^{8} d^{7} - 21 \, b^{2} c^{7} d^{6} e + 21 \, b^{3} c^{6} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{3} e^{4} + 42 \, b^{6} c^{3} d^{2} e^{5} - 14 \, b^{7} c^{2} d e^{6} + b^{8} c e^{7}\right )} x^{3} +{\left (18 \, b^{2} c^{7} d^{7} - 63 \, b^{3} c^{6} d^{6} e + 63 \, b^{4} c^{5} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{4} e^{3} - 35 \, b^{6} c^{3} d^{3} e^{4} + 63 \, b^{7} c^{2} d^{2} e^{5} - 35 \, b^{8} c d e^{6} + 7 \, b^{9} e^{7}\right )} x^{2} + 2 \,{\left (2 \, b^{3} c^{6} d^{7} - 7 \, b^{4} c^{5} d^{6} e\right )} x - 6 \,{\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2} - 7 \, b^{5} c^{4} d^{2} e^{5} + 7 \, b^{6} c^{3} d e^{6} - 2 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \,{\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2} - 7 \, b^{6} c^{3} d^{2} e^{5} + 7 \, b^{7} c^{2} d e^{6} - 2 \, b^{8} c e^{7}\right )} x^{3} +{\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2} - 7 \, b^{7} c^{2} d^{2} e^{5} + 7 \, b^{8} c d e^{6} - 2 \, b^{9} e^{7}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{7} x^{4} + 2 \, b^{6} c^{6} x^{3} + b^{7} c^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/2*(b^5*c^4*e^7*x^6 - b^4*c^5*d^7 + 2*(7*b^5*c^4*d*e^6 - 2*b^6*c^3*e^7)*x^5 + (28*b^6*c^3*d*e^6 - 11*b^7*c^2*
e^7)*x^4 + 2*(6*b*c^8*d^7 - 21*b^2*c^7*d^6*e + 21*b^3*c^6*d^5*e^2 - 35*b^5*c^4*d^3*e^4 + 42*b^6*c^3*d^2*e^5 -
14*b^7*c^2*d*e^6 + b^8*c*e^7)*x^3 + (18*b^2*c^7*d^7 - 63*b^3*c^6*d^6*e + 63*b^4*c^5*d^5*e^2 - 35*b^5*c^4*d^4*e
^3 - 35*b^6*c^3*d^3*e^4 + 63*b^7*c^2*d^2*e^5 - 35*b^8*c*d*e^6 + 7*b^9*e^7)*x^2 + 2*(2*b^3*c^6*d^7 - 7*b^4*c^5*
d^6*e)*x - 6*((2*c^9*d^7 - 7*b*c^8*d^6*e + 7*b^2*c^7*d^5*e^2 - 7*b^5*c^4*d^2*e^5 + 7*b^6*c^3*d*e^6 - 2*b^7*c^2
*e^7)*x^4 + 2*(2*b*c^8*d^7 - 7*b^2*c^7*d^6*e + 7*b^3*c^6*d^5*e^2 - 7*b^6*c^3*d^2*e^5 + 7*b^7*c^2*d*e^6 - 2*b^8
*c*e^7)*x^3 + (2*b^2*c^7*d^7 - 7*b^3*c^6*d^6*e + 7*b^4*c^5*d^5*e^2 - 7*b^7*c^2*d^2*e^5 + 7*b^8*c*d*e^6 - 2*b^9
*e^7)*x^2)*log(c*x + b) + 6*((2*c^9*d^7 - 7*b*c^8*d^6*e + 7*b^2*c^7*d^5*e^2)*x^4 + 2*(2*b*c^8*d^7 - 7*b^2*c^7*
d^6*e + 7*b^3*c^6*d^5*e^2)*x^3 + (2*b^2*c^7*d^7 - 7*b^3*c^6*d^6*e + 7*b^4*c^5*d^5*e^2)*x^2)*log(x))/(b^5*c^7*x
^4 + 2*b^6*c^6*x^3 + b^7*c^5*x^2)

________________________________________________________________________________________

Sympy [B]  time = 61.7282, size = 685, normalized size = 3.37 \begin{align*} \frac{- b^{3} c^{5} d^{7} + x^{3} \left (8 b^{7} c e^{7} - 42 b^{6} c^{2} d e^{6} + 84 b^{5} c^{3} d^{2} e^{5} - 70 b^{4} c^{4} d^{3} e^{4} + 42 b^{2} c^{6} d^{5} e^{2} - 42 b c^{7} d^{6} e + 12 c^{8} d^{7}\right ) + x^{2} \left (7 b^{8} e^{7} - 35 b^{7} c d e^{6} + 63 b^{6} c^{2} d^{2} e^{5} - 35 b^{5} c^{3} d^{3} e^{4} - 35 b^{4} c^{4} d^{4} e^{3} + 63 b^{3} c^{5} d^{5} e^{2} - 63 b^{2} c^{6} d^{6} e + 18 b c^{7} d^{7}\right ) + x \left (- 14 b^{3} c^{5} d^{6} e + 4 b^{2} c^{6} d^{7}\right )}{2 b^{6} c^{5} x^{2} + 4 b^{5} c^{6} x^{3} + 2 b^{4} c^{7} x^{4}} + \frac{e^{7} x^{2}}{2 c^{3}} - \frac{x \left (3 b e^{7} - 7 c d e^{6}\right )}{c^{4}} + \frac{3 d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + 3 b c^{4} d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right )}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5}} + \frac{3 \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + \frac{3 b \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right )}{c}}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5} c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**7/(c*x**2+b*x)**3,x)

[Out]

(-b**3*c**5*d**7 + x**3*(8*b**7*c*e**7 - 42*b**6*c**2*d*e**6 + 84*b**5*c**3*d**2*e**5 - 70*b**4*c**4*d**3*e**4
+ 42*b**2*c**6*d**5*e**2 - 42*b*c**7*d**6*e + 12*c**8*d**7) + x**2*(7*b**8*e**7 - 35*b**7*c*d*e**6 + 63*b**6*
c**2*d**2*e**5 - 35*b**5*c**3*d**3*e**4 - 35*b**4*c**4*d**4*e**3 + 63*b**3*c**5*d**5*e**2 - 63*b**2*c**6*d**6*
e + 18*b*c**7*d**7) + x*(-14*b**3*c**5*d**6*e + 4*b**2*c**6*d**7))/(2*b**6*c**5*x**2 + 4*b**5*c**6*x**3 + 2*b*
*4*c**7*x**4) + e**7*x**2/(2*c**3) - x*(3*b*e**7 - 7*c*d*e**6)/c**4 + 3*d**5*(7*b**2*e**2 - 7*b*c*d*e + 2*c**2
*d**2)*log(x + (-21*b**3*c**4*d**5*e**2 + 21*b**2*c**5*d**6*e - 6*b*c**6*d**7 + 3*b*c**4*d**5*(7*b**2*e**2 - 7
*b*c*d*e + 2*c**2*d**2))/(6*b**7*e**7 - 21*b**6*c*d*e**6 + 21*b**5*c**2*d**2*e**5 - 42*b**2*c**5*d**5*e**2 + 4
2*b*c**6*d**6*e - 12*c**7*d**7))/b**5 + 3*(b*e - c*d)**5*(2*b**2*e**2 + 3*b*c*d*e + 2*c**2*d**2)*log(x + (-21*
b**3*c**4*d**5*e**2 + 21*b**2*c**5*d**6*e - 6*b*c**6*d**7 + 3*b*(b*e - c*d)**5*(2*b**2*e**2 + 3*b*c*d*e + 2*c*
*2*d**2)/c)/(6*b**7*e**7 - 21*b**6*c*d*e**6 + 21*b**5*c**2*d**2*e**5 - 42*b**2*c**5*d**5*e**2 + 42*b*c**6*d**6
*e - 12*c**7*d**7))/(b**5*c**5)

________________________________________________________________________________________

Giac [A]  time = 2.01136, size = 517, normalized size = 2.55 \begin{align*} \frac{3 \,{\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{c^{3} x^{2} e^{7} + 14 \, c^{3} d x e^{6} - 6 \, b c^{2} x e^{7}}{2 \, c^{6}} - \frac{3 \,{\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{5}} - \frac{b^{3} c^{5} d^{7} - 2 \,{\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} -{\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

3*(2*c^2*d^7 - 7*b*c*d^6*e + 7*b^2*d^5*e^2)*log(abs(x))/b^5 + 1/2*(c^3*x^2*e^7 + 14*c^3*d*x*e^6 - 6*b*c^2*x*e^
7)/c^6 - 3*(2*c^7*d^7 - 7*b*c^6*d^6*e + 7*b^2*c^5*d^5*e^2 - 7*b^5*c^2*d^2*e^5 + 7*b^6*c*d*e^6 - 2*b^7*e^7)*log
(abs(c*x + b))/(b^5*c^5) - 1/2*(b^3*c^5*d^7 - 2*(6*c^8*d^7 - 21*b*c^7*d^6*e + 21*b^2*c^6*d^5*e^2 - 35*b^4*c^4*
d^3*e^4 + 42*b^5*c^3*d^2*e^5 - 21*b^6*c^2*d*e^6 + 4*b^7*c*e^7)*x^3 - (18*b*c^7*d^7 - 63*b^2*c^6*d^6*e + 63*b^3
*c^5*d^5*e^2 - 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 63*b^6*c^2*d^2*e^5 - 35*b^7*c*d*e^6 + 7*b^8*e^7)*x^2
- 2*(2*b^2*c^6*d^7 - 7*b^3*c^5*d^6*e)*x)/((c*x + b)^2*b^4*c^5*x^2)