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

Optimal. Leaf size=94 \[ \frac{b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac{b^2 (3 c d-4 b e)}{c^5 (b+c x)}-\frac{3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac{x (c d-3 b e)}{c^4}+\frac{e x^2}{2 c^3} \]

[Out]

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

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Rubi [A]  time = 0.220949, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac{b^2 (3 c d-4 b e)}{c^5 (b+c x)}-\frac{3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac{x (c d-3 b e)}{c^4}+\frac{e x^2}{2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.103666, size = 86, normalized size = 0.91 \[ \frac{\frac{b^3 (c d-b e)}{(b+c x)^2}+\frac{2 b^2 (4 b e-3 c d)}{b+c x}+2 c x (c d-3 b e)+6 b (2 b e-c d) \log (b+c x)+c^2 e x^2}{2 c^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 117, normalized size = 1.2 \[{\frac{e{x}^{2}}{2\,{c}^{3}}}-3\,{\frac{bex}{{c}^{4}}}+{\frac{dx}{{c}^{3}}}+6\,{\frac{{b}^{2}\ln \left ( cx+b \right ) e}{{c}^{5}}}-3\,{\frac{b\ln \left ( cx+b \right ) d}{{c}^{4}}}-{\frac{{b}^{4}e}{2\,{c}^{5} \left ( cx+b \right ) ^{2}}}+{\frac{d{b}^{3}}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}}+4\,{\frac{{b}^{3}e}{{c}^{5} \left ( cx+b \right ) }}-3\,{\frac{{b}^{2}d}{{c}^{4} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.696695, size = 143, normalized size = 1.52 \[ -\frac{5 \, b^{3} c d - 7 \, b^{4} e + 2 \,{\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x}{2 \,{\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} + \frac{c e x^{2} + 2 \,{\left (c d - 3 \, b e\right )} x}{2 \, c^{4}} - \frac{3 \,{\left (b c d - 2 \, b^{2} e\right )} \log \left (c x + b\right )}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272734, size = 225, normalized size = 2.39 \[ \frac{c^{4} e x^{4} - 5 \, b^{3} c d + 7 \, b^{4} e + 2 \,{\left (c^{4} d - 2 \, b c^{3} e\right )} x^{3} +{\left (4 \, b c^{3} d - 11 \, b^{2} c^{2} e\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x - 6 \,{\left (b^{3} c d - 2 \, b^{4} e +{\left (b c^{3} d - 2 \, b^{2} c^{2} e\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - 2 \, b^{3} c e\right )} x\right )} \log \left (c x + b\right )}{2 \,{\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.37025, size = 105, normalized size = 1.12 \[ \frac{3 b \left (2 b e - c d\right ) \log{\left (b + c x \right )}}{c^{5}} + \frac{7 b^{4} e - 5 b^{3} c d + x \left (8 b^{3} c e - 6 b^{2} c^{2} d\right )}{2 b^{2} c^{5} + 4 b c^{6} x + 2 c^{7} x^{2}} + \frac{e x^{2}}{2 c^{3}} - \frac{x \left (3 b e - c d\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b*(2*b*e - c*d)*log(b + c*x)/c**5 + (7*b**4*e - 5*b**3*c*d + x*(8*b**3*c*e - 6
*b**2*c**2*d))/(2*b**2*c**5 + 4*b*c**6*x + 2*c**7*x**2) + e*x**2/(2*c**3) - x*(3
*b*e - c*d)/c**4

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GIAC/XCAS [A]  time = 0.268616, size = 140, normalized size = 1.49 \[ -\frac{3 \,{\left (b c d - 2 \, b^{2} e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{c^{5}} + \frac{c^{3} x^{2} e + 2 \, c^{3} d x - 6 \, b c^{2} x e}{2 \, c^{6}} - \frac{5 \, b^{3} c d - 7 \, b^{4} e + 2 \,{\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x}{2 \,{\left (c x + b\right )}^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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