3.276 \(\int \frac{(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=179 \[ \frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{3 d^5 (c d-2 b e)}{b^4 x}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac{d^6}{2 b^3 x^2}+\frac{3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac{e^6 x}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.501354, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{3 d^5 (c d-2 b e)}{b^4 x}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac{d^6}{2 b^3 x^2}+\frac{3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac{e^6 x}{c^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(b*x + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{6} \int \frac{1}{c^{3}}\, dx - \frac{d^{6}}{2 b^{3} x^{2}} + \frac{\left (b e - c d\right )^{6}}{2 b^{3} c^{4} \left (b + c x\right )^{2}} - \frac{3 d^{5} \left (2 b e - c d\right )}{b^{4} x} - \frac{3 \left (b e - c d\right )^{5} \left (b e + c d\right )}{b^{4} c^{4} \left (b + c x\right )} + \frac{3 d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x \right )}}{b^{5}} - \frac{3 \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(c*x**2+b*x)**3,x)

[Out]

e**6*Integral(c**(-3), x) - d**6/(2*b**3*x**2) + (b*e - c*d)**6/(2*b**3*c**4*(b
+ c*x)**2) - 3*d**5*(2*b*e - c*d)/(b**4*x) - 3*(b*e - c*d)**5*(b*e + c*d)/(b**4*
c**4*(b + c*x)) + 3*d**4*(5*b**2*e**2 - 6*b*c*d*e + 2*c**2*d**2)*log(x)/b**5 - 3
*(b*e - c*d)**4*(b**2*e**2 + 2*b*c*d*e + 2*c**2*d**2)*log(b + c*x)/(b**5*c**4)

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Mathematica [A]  time = 0.148395, size = 179, normalized size = 1. \[ \frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{3 d^5 (c d-2 b e)}{b^4 x}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac{d^6}{2 b^3 x^2}+\frac{3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac{e^6 x}{c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.023, size = 396, normalized size = 2.2 \[ -12\,{\frac{c{d}^{5}e}{{b}^{3} \left ( cx+b \right ) }}-18\,{\frac{{d}^{5}\ln \left ( x \right ) ce}{{b}^{4}}}-{\frac{{d}^{6}}{2\,{b}^{3}{x}^{2}}}+18\,{\frac{c\ln \left ( cx+b \right ){d}^{5}e}{{b}^{4}}}-3\,{\frac{{b}^{2}d{e}^{5}}{{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{15\,b{d}^{2}{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{e}^{6}x}{{c}^{3}}}-3\,{\frac{c{d}^{5}e}{{b}^{2} \left ( cx+b \right ) ^{2}}}+12\,{\frac{bd{e}^{5}}{{c}^{3} \left ( cx+b \right ) }}-10\,{\frac{{d}^{3}{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+{\frac{15\,{d}^{4}{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}+15\,{\frac{{d}^{4}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{6}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-6\,{\frac{{d}^{5}e}{{b}^{3}x}}+3\,{\frac{{d}^{6}c}{{b}^{4}x}}-3\,{\frac{b\ln \left ( cx+b \right ){e}^{6}}{{c}^{4}}}+6\,{\frac{\ln \left ( cx+b \right ) d{e}^{5}}{{c}^{3}}}-15\,{\frac{\ln \left ( cx+b \right ){d}^{4}{e}^{2}}{{b}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ){d}^{6}}{{b}^{5}}}-3\,{\frac{{b}^{2}{e}^{6}}{{c}^{4} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{6}}{{b}^{4} \left ( cx+b \right ) }}-15\,{\frac{{d}^{2}{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+15\,{\frac{{d}^{4}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}+{\frac{{b}^{3}{e}^{6}}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{6}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(c*x^2+b*x)^3,x)

[Out]

-12*c/b^3/(c*x+b)*d^5*e-18*d^5/b^4*ln(x)*c*e-1/2*d^6/b^3/x^2+18*c/b^4*ln(c*x+b)*
d^5*e-3/c^3*b^2/(c*x+b)^2*d*e^5+15/2/c^2*b/(c*x+b)^2*d^2*e^4+e^6*x/c^3-3*c/b^2/(
c*x+b)^2*d^5*e+12/c^3*b/(c*x+b)*d*e^5-10/c/(c*x+b)^2*d^3*e^3+15/2/b/(c*x+b)^2*d^
4*e^2+15*d^4/b^3*ln(x)*e^2+6*d^6/b^5*ln(x)*c^2-6*d^5/b^3/x*e+3*d^6/b^4/x*c-3/c^4
*b*ln(c*x+b)*e^6+6/c^3*ln(c*x+b)*d*e^5-15/b^3*ln(c*x+b)*d^4*e^2-6*c^2/b^5*ln(c*x
+b)*d^6-3/c^4*b^2/(c*x+b)*e^6+3*c^2/b^4/(c*x+b)*d^6-15/c^2/(c*x+b)*d^2*e^4+15/b^
2/(c*x+b)*d^4*e^2+1/2/c^4*b^3/(c*x+b)^2*e^6+1/2*c^2/b^3/(c*x+b)^2*d^6

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Maxima [A]  time = 0.709606, size = 460, normalized size = 2.57 \[ \frac{e^{6} x}{c^{3}} - \frac{b^{3} c^{4} d^{6} - 6 \,{\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} -{\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \,{\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \,{\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} + \frac{3 \,{\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left (c x + b\right )}{b^{5} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

e^6*x/c^3 - 1/2*(b^3*c^4*d^6 - 6*(2*c^7*d^6 - 6*b*c^6*d^5*e + 5*b^2*c^5*d^4*e^2
- 5*b^4*c^3*d^2*e^4 + 4*b^5*c^2*d*e^5 - b^6*c*e^6)*x^3 - (18*b*c^6*d^6 - 54*b^2*
c^5*d^5*e + 45*b^3*c^4*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 15*b^5*c^2*d^2*e^4 + 18*b^
6*c*d*e^5 - 5*b^7*e^6)*x^2 - 4*(b^2*c^5*d^6 - 3*b^3*c^4*d^5*e)*x)/(b^4*c^6*x^4 +
2*b^5*c^5*x^3 + b^6*c^4*x^2) + 3*(2*c^2*d^6 - 6*b*c*d^5*e + 5*b^2*d^4*e^2)*log(
x)/b^5 - 3*(2*c^6*d^6 - 6*b*c^5*d^5*e + 5*b^2*c^4*d^4*e^2 - 2*b^5*c*d*e^5 + b^6*
e^6)*log(c*x + b)/(b^5*c^4)

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Fricas [A]  time = 0.251461, size = 782, normalized size = 4.37 \[ \frac{2 \, b^{5} c^{3} e^{6} x^{5} + 4 \, b^{6} c^{2} e^{6} x^{4} - b^{4} c^{4} d^{6} + 2 \,{\left (6 \, b c^{7} d^{6} - 18 \, b^{2} c^{6} d^{5} e + 15 \, b^{3} c^{5} d^{4} e^{2} - 15 \, b^{5} c^{3} d^{2} e^{4} + 12 \, b^{6} c^{2} d e^{5} - 2 \, b^{7} c e^{6}\right )} x^{3} +{\left (18 \, b^{2} c^{6} d^{6} - 54 \, b^{3} c^{5} d^{5} e + 45 \, b^{4} c^{4} d^{4} e^{2} - 20 \, b^{5} c^{3} d^{3} e^{3} - 15 \, b^{6} c^{2} d^{2} e^{4} + 18 \, b^{7} c d e^{5} - 5 \, b^{8} e^{6}\right )} x^{2} + 4 \,{\left (b^{3} c^{5} d^{6} - 3 \, b^{4} c^{4} d^{5} e\right )} x - 6 \,{\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2} - 2 \, b^{5} c^{3} d e^{5} + b^{6} c^{2} e^{6}\right )} x^{4} + 2 \,{\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2} - 2 \, b^{6} c^{2} d e^{5} + b^{7} c e^{6}\right )} x^{3} +{\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2} - 2 \, b^{7} c d e^{5} + b^{8} e^{6}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{6} x^{4} + 2 \, b^{6} c^{5} x^{3} + b^{7} c^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

1/2*(2*b^5*c^3*e^6*x^5 + 4*b^6*c^2*e^6*x^4 - b^4*c^4*d^6 + 2*(6*b*c^7*d^6 - 18*b
^2*c^6*d^5*e + 15*b^3*c^5*d^4*e^2 - 15*b^5*c^3*d^2*e^4 + 12*b^6*c^2*d*e^5 - 2*b^
7*c*e^6)*x^3 + (18*b^2*c^6*d^6 - 54*b^3*c^5*d^5*e + 45*b^4*c^4*d^4*e^2 - 20*b^5*
c^3*d^3*e^3 - 15*b^6*c^2*d^2*e^4 + 18*b^7*c*d*e^5 - 5*b^8*e^6)*x^2 + 4*(b^3*c^5*
d^6 - 3*b^4*c^4*d^5*e)*x - 6*((2*c^8*d^6 - 6*b*c^7*d^5*e + 5*b^2*c^6*d^4*e^2 - 2
*b^5*c^3*d*e^5 + b^6*c^2*e^6)*x^4 + 2*(2*b*c^7*d^6 - 6*b^2*c^6*d^5*e + 5*b^3*c^5
*d^4*e^2 - 2*b^6*c^2*d*e^5 + b^7*c*e^6)*x^3 + (2*b^2*c^6*d^6 - 6*b^3*c^5*d^5*e +
5*b^4*c^4*d^4*e^2 - 2*b^7*c*d*e^5 + b^8*e^6)*x^2)*log(c*x + b) + 6*((2*c^8*d^6
- 6*b*c^7*d^5*e + 5*b^2*c^6*d^4*e^2)*x^4 + 2*(2*b*c^7*d^6 - 6*b^2*c^6*d^5*e + 5*
b^3*c^5*d^4*e^2)*x^3 + (2*b^2*c^6*d^6 - 6*b^3*c^5*d^5*e + 5*b^4*c^4*d^4*e^2)*x^2
)*log(x))/(b^5*c^6*x^4 + 2*b^6*c^5*x^3 + b^7*c^4*x^2)

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Sympy [A]  time = 46.7312, size = 597, normalized size = 3.34 \[ - \frac{b^{3} c^{4} d^{6} + x^{3} \left (6 b^{6} c e^{6} - 24 b^{5} c^{2} d e^{5} + 30 b^{4} c^{3} d^{2} e^{4} - 30 b^{2} c^{5} d^{4} e^{2} + 36 b c^{6} d^{5} e - 12 c^{7} d^{6}\right ) + x^{2} \left (5 b^{7} e^{6} - 18 b^{6} c d e^{5} + 15 b^{5} c^{2} d^{2} e^{4} + 20 b^{4} c^{3} d^{3} e^{3} - 45 b^{3} c^{4} d^{4} e^{2} + 54 b^{2} c^{5} d^{5} e - 18 b c^{6} d^{6}\right ) + x \left (12 b^{3} c^{4} d^{5} e - 4 b^{2} c^{5} d^{6}\right )}{2 b^{6} c^{4} x^{2} + 4 b^{5} c^{5} x^{3} + 2 b^{4} c^{6} x^{4}} + \frac{e^{6} x}{c^{3}} + \frac{3 d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} - 3 b c^{3} d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right )}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5}} - \frac{3 \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} + \frac{3 b \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right )}{c}}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(c*x**2+b*x)**3,x)

[Out]

-(b**3*c**4*d**6 + x**3*(6*b**6*c*e**6 - 24*b**5*c**2*d*e**5 + 30*b**4*c**3*d**2
*e**4 - 30*b**2*c**5*d**4*e**2 + 36*b*c**6*d**5*e - 12*c**7*d**6) + x**2*(5*b**7
*e**6 - 18*b**6*c*d*e**5 + 15*b**5*c**2*d**2*e**4 + 20*b**4*c**3*d**3*e**3 - 45*
b**3*c**4*d**4*e**2 + 54*b**2*c**5*d**5*e - 18*b*c**6*d**6) + x*(12*b**3*c**4*d*
*5*e - 4*b**2*c**5*d**6))/(2*b**6*c**4*x**2 + 4*b**5*c**5*x**3 + 2*b**4*c**6*x**
4) + e**6*x/c**3 + 3*d**4*(5*b**2*e**2 - 6*b*c*d*e + 2*c**2*d**2)*log(x + (15*b*
*3*c**3*d**4*e**2 - 18*b**2*c**4*d**5*e + 6*b*c**5*d**6 - 3*b*c**3*d**4*(5*b**2*
e**2 - 6*b*c*d*e + 2*c**2*d**2))/(3*b**6*e**6 - 6*b**5*c*d*e**5 + 30*b**2*c**4*d
**4*e**2 - 36*b*c**5*d**5*e + 12*c**6*d**6))/b**5 - 3*(b*e - c*d)**4*(b**2*e**2
+ 2*b*c*d*e + 2*c**2*d**2)*log(x + (15*b**3*c**3*d**4*e**2 - 18*b**2*c**4*d**5*e
+ 6*b*c**5*d**6 + 3*b*(b*e - c*d)**4*(b**2*e**2 + 2*b*c*d*e + 2*c**2*d**2)/c)/(
3*b**6*e**6 - 6*b**5*c*d*e**5 + 30*b**2*c**4*d**4*e**2 - 36*b*c**5*d**5*e + 12*c
**6*d**6))/(b**5*c**4)

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GIAC/XCAS [A]  time = 0.2079, size = 427, normalized size = 2.39 \[ \frac{x e^{6}}{c^{3}} + \frac{3 \,{\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c^{4}} - \frac{b^{3} c^{4} d^{6} - 6 \,{\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} -{\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \,{\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

x*e^6/c^3 + 3*(2*c^2*d^6 - 6*b*c*d^5*e + 5*b^2*d^4*e^2)*ln(abs(x))/b^5 - 3*(2*c^
6*d^6 - 6*b*c^5*d^5*e + 5*b^2*c^4*d^4*e^2 - 2*b^5*c*d*e^5 + b^6*e^6)*ln(abs(c*x
+ b))/(b^5*c^4) - 1/2*(b^3*c^4*d^6 - 6*(2*c^7*d^6 - 6*b*c^6*d^5*e + 5*b^2*c^5*d^
4*e^2 - 5*b^4*c^3*d^2*e^4 + 4*b^5*c^2*d*e^5 - b^6*c*e^6)*x^3 - (18*b*c^6*d^6 - 5
4*b^2*c^5*d^5*e + 45*b^3*c^4*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 15*b^5*c^2*d^2*e^4 +
18*b^6*c*d*e^5 - 5*b^7*e^6)*x^2 - 4*(b^2*c^5*d^6 - 3*b^3*c^4*d^5*e)*x)/((c*x +
b)^2*b^4*c^4*x^2)