3.1.51 \(\int \frac {x^5 (d+e x)}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=90 \[ \frac {b^3 (c d-b e)}{c^5 (b+c x)}+\frac {b^2 (3 c d-4 b e) \log (b+c x)}{c^5}-\frac {b x (2 c d-3 b e)}{c^4}+\frac {x^2 (c d-2 b e)}{2 c^3}+\frac {e x^3}{3 c^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 90, 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)}{c^5 (b+c x)}+\frac {b^2 (3 c d-4 b e) \log (b+c x)}{c^5}+\frac {x^2 (c d-2 b e)}{2 c^3}-\frac {b x (2 c d-3 b e)}{c^4}+\frac {e x^3}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 87, normalized size = 0.97 \begin {gather*} \frac {\frac {6 b^3 (c d-b e)}{b+c x}+6 b^2 (3 c d-4 b e) \log (b+c x)+3 c^2 x^2 (c d-2 b e)+6 b c x (3 b e-2 c d)+2 c^3 e x^3}{6 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 (d+e x)}{\left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^5*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(x^5*(d + e*x))/(b*x + c*x^2)^2, x]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 139, normalized size = 1.54 \begin {gather*} \frac {2 \, c^{4} e x^{4} + 6 \, b^{3} c d - 6 \, b^{4} e + {\left (3 \, c^{4} d - 4 \, b c^{3} e\right )} x^{3} - 3 \, {\left (3 \, b c^{3} d - 4 \, b^{2} c^{2} e\right )} x^{2} - 6 \, {\left (2 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x + 6 \, {\left (3 \, b^{3} c d - 4 \, b^{4} e + {\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x\right )} \log \left (c x + b\right )}{6 \, {\left (c^{6} x + b c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 107, normalized size = 1.19 \begin {gather*} \frac {{\left (3 \, b^{2} c d - 4 \, b^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} + \frac {2 \, c^{4} x^{3} e + 3 \, c^{4} d x^{2} - 6 \, b c^{3} x^{2} e - 12 \, b c^{3} d x + 18 \, b^{2} c^{2} x e}{6 \, c^{6}} + \frac {b^{3} c d - b^{4} e}{{\left (c x + b\right )} c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 109, normalized size = 1.21 \begin {gather*} \frac {e \,x^{3}}{3 c^{2}}-\frac {b e \,x^{2}}{c^{3}}+\frac {d \,x^{2}}{2 c^{2}}-\frac {b^{4} e}{\left (c x +b \right ) c^{5}}+\frac {b^{3} d}{\left (c x +b \right ) c^{4}}-\frac {4 b^{3} e \ln \left (c x +b \right )}{c^{5}}+\frac {3 b^{2} d \ln \left (c x +b \right )}{c^{4}}+\frac {3 b^{2} e x}{c^{4}}-\frac {2 b d x}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x+d)/(c*x^2+b*x)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.93, size = 98, normalized size = 1.09 \begin {gather*} \frac {b^{3} c d - b^{4} e}{c^{6} x + b c^{5}} + \frac {2 \, c^{2} e x^{3} + 3 \, {\left (c^{2} d - 2 \, b c e\right )} x^{2} - 6 \, {\left (2 \, b c d - 3 \, b^{2} e\right )} x}{6 \, c^{4}} + \frac {{\left (3 \, b^{2} c d - 4 \, b^{3} 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^5*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.03, size = 115, normalized size = 1.28 \begin {gather*} x^2\,\left (\frac {d}{2\,c^2}-\frac {b\,e}{c^3}\right )-x\,\left (\frac {b^2\,e}{c^4}+\frac {2\,b\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )}{c}\right )-\frac {\ln \left (b+c\,x\right )\,\left (4\,b^3\,e-3\,b^2\,c\,d\right )}{c^5}+\frac {e\,x^3}{3\,c^2}-\frac {b^4\,e-b^3\,c\,d}{c\,\left (x\,c^5+b\,c^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(d + e*x))/(b*x + c*x^2)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.42, size = 92, normalized size = 1.02 \begin {gather*} - \frac {b^{2} \left (4 b e - 3 c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x^{2} \left (- \frac {b e}{c^{3}} + \frac {d}{2 c^{2}}\right ) + x \left (\frac {3 b^{2} e}{c^{4}} - \frac {2 b d}{c^{3}}\right ) + \frac {- b^{4} e + b^{3} c d}{b c^{5} + c^{6} x} + \frac {e x^{3}}{3 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x+d)/(c*x**2+b*x)**2,x)

[Out]

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

________________________________________________________________________________________