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

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

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Rubi [A]  time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \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 {x (c d-3 b e)}{c^4}-\frac {3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac {e x^2}{2 c^3} \end {gather*}

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 86, normalized size = 0.91 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 167, normalized size = 1.78 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)/(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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giac [A]  time = 0.15, size = 104, normalized size = 1.11 \begin {gather*} -\frac {3 \, {\left (b c d - 2 \, b^{2} e\right )} \log \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(b*c*d - 2*b^2*e)*log(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)

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maple [A]  time = 0.07, size = 117, normalized size = 1.24 \begin {gather*} -\frac {b^{4} e}{2 \left (c x +b \right )^{2} c^{5}}+\frac {b^{3} d}{2 \left (c x +b \right )^{2} c^{4}}+\frac {e \,x^{2}}{2 c^{3}}+\frac {4 b^{3} e}{\left (c x +b \right ) c^{5}}-\frac {3 b^{2} d}{\left (c x +b \right ) c^{4}}+\frac {6 b^{2} e \ln \left (c x +b \right )}{c^{5}}-\frac {3 b d \ln \left (c x +b \right )}{c^{4}}-\frac {3 b e x}{c^{4}}+\frac {d x}{c^{3}} \end {gather*}

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*x*b*e+1/c^3*x*d+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 = 0.81, size = 106, normalized size = 1.13 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)/(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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mupad [B]  time = 1.04, size = 108, normalized size = 1.15 \begin {gather*} x\,\left (\frac {d}{c^3}-\frac {3\,b\,e}{c^4}\right )+\frac {x\,\left (4\,b^3\,e-3\,b^2\,c\,d\right )+\frac {7\,b^4\,e-5\,b^3\,c\,d}{2\,c}}{b^2\,c^4+2\,b\,c^5\,x+c^6\,x^2}+\frac {e\,x^2}{2\,c^3}+\frac {\ln \left (b+c\,x\right )\,\left (6\,b^2\,e-3\,b\,c\,d\right )}{c^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.68, size = 107, normalized size = 1.14 \begin {gather*} \frac {3 b \left (2 b e - c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x \left (- \frac {3 b e}{c^{4}} + \frac {d}{c^{3}}\right ) + \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}} \end {gather*}

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 + x*(-3*b*e/c**4 + d/c**3) + (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)

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