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

Optimal. Leaf size=131 \[ -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac {d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac {c^2 \log (d+e x)}{e^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 131, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac {d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac {c^2 \log (d+e x)}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^5,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 126, normalized size = 0.96 \begin {gather*} \frac {-b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )-6 b c e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) - 6*b*c*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + c^2*d*(25*d^3
+ 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 225, normalized size = 1.72 \begin {gather*} \frac {25 \, c^{2} d^{4} - 6 \, b c d^{3} e - b^{2} d^{2} e^{2} + 24 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \, {\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - b^{2} e^{4}\right )} x^{2} + 4 \, {\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - b^{2} d e^{3}\right )} x + 12 \, {\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - b^2*d^2*e^2 + 24*(2*c^2*d*e^3 - b*c*e^4)*x^3 + 6*(18*c^2*d^2*e^2 - 6*b*c*d*e^
3 - b^2*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - b^2*d*e^3)*x + 12*(c^2*e^4*x^4 + 4*c^2*d*e^3*x^3 + 6*c^2*
d^2*e^2*x^2 + 4*c^2*d^3*e*x + c^2*d^4)*log(e*x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^
4*e^5)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 214, normalized size = 1.63 \begin {gather*} -c^{2} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{12} \, {\left (\frac {48 \, c^{2} d e^{15}}{x e + d} - \frac {36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac {16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac {3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac {24 \, b c e^{16}}{x e + d} + \frac {36 \, b c d e^{16}}{{\left (x e + d\right )}^{2}} - \frac {24 \, b c d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac {6 \, b c d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac {6 \, b^{2} e^{17}}{{\left (x e + d\right )}^{2}} + \frac {8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac {3 \, b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^2*e^(-5)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*c^2*d*e^15/(x*e + d) - 36*c^2*d^2*e^15/(x*e + d)^2
+ 16*c^2*d^3*e^15/(x*e + d)^3 - 3*c^2*d^4*e^15/(x*e + d)^4 - 24*b*c*e^16/(x*e + d) + 36*b*c*d*e^16/(x*e + d)^
2 - 24*b*c*d^2*e^16/(x*e + d)^3 + 6*b*c*d^3*e^16/(x*e + d)^4 - 6*b^2*e^17/(x*e + d)^2 + 8*b^2*d*e^17/(x*e + d)
^3 - 3*b^2*d^2*e^17/(x*e + d)^4)*e^(-20)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 197, normalized size = 1.50 \begin {gather*} -\frac {b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {b c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {2 b^{2} d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 b c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 b c d}{\left (e x +d \right )^{2} e^{4}}-\frac {3 c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}-\frac {2 b c}{\left (e x +d \right ) e^{4}}+\frac {4 c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.42, size = 177, normalized size = 1.35 \begin {gather*} \frac {25 \, c^{2} d^{4} - 6 \, b c d^{3} e - b^{2} d^{2} e^{2} + 24 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \, {\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - b^{2} e^{4}\right )} x^{2} + 4 \, {\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - b^{2} d e^{3}\right )} x}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac {c^{2} \log \left (e x + d\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - b^2*d^2*e^2 + 24*(2*c^2*d*e^3 - b*c*e^4)*x^3 + 6*(18*c^2*d^2*e^2 - 6*b*c*d*e^
3 - b^2*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - b^2*d*e^3)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*
d^3*e^6*x + d^4*e^5) + c^2*log(e*x + d)/e^5

________________________________________________________________________________________

mupad [B]  time = 0.23, size = 167, normalized size = 1.27 \begin {gather*} \frac {c^2\,\ln \left (d+e\,x\right )}{e^5}-\frac {\frac {b^2\,d^2\,e^2+6\,b\,c\,d^3\,e-25\,c^2\,d^4}{12\,e^5}+\frac {x^2\,\left (b^2\,e^2+6\,b\,c\,d\,e-18\,c^2\,d^2\right )}{2\,e^3}+\frac {x\,\left (b^2\,d\,e^2+6\,b\,c\,d^2\,e-22\,c^2\,d^3\right )}{3\,e^4}+\frac {2\,c\,x^3\,\left (b\,e-2\,c\,d\right )}{e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2/(d + e*x)^5,x)

[Out]

(c^2*log(d + e*x))/e^5 - ((b^2*d^2*e^2 - 25*c^2*d^4 + 6*b*c*d^3*e)/(12*e^5) + (x^2*(b^2*e^2 - 18*c^2*d^2 + 6*b
*c*d*e))/(2*e^3) + (x*(b^2*d*e^2 - 22*c^2*d^3 + 6*b*c*d^2*e))/(3*e^4) + (2*c*x^3*(b*e - 2*c*d))/e^2)/(d^4 + e^
4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x)

________________________________________________________________________________________

sympy [A]  time = 1.88, size = 180, normalized size = 1.37 \begin {gather*} \frac {c^{2} \log {\left (d + e x \right )}}{e^{5}} + \frac {- b^{2} d^{2} e^{2} - 6 b c d^{3} e + 25 c^{2} d^{4} + x^{3} \left (- 24 b c e^{4} + 48 c^{2} d e^{3}\right ) + x^{2} \left (- 6 b^{2} e^{4} - 36 b c d e^{3} + 108 c^{2} d^{2} e^{2}\right ) + x \left (- 4 b^{2} d e^{3} - 24 b c d^{2} e^{2} + 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*log(d + e*x)/e**5 + (-b**2*d**2*e**2 - 6*b*c*d**3*e + 25*c**2*d**4 + x**3*(-24*b*c*e**4 + 48*c**2*d*e**3)
+ x**2*(-6*b**2*e**4 - 36*b*c*d*e**3 + 108*c**2*d**2*e**2) + x*(-4*b**2*d*e**3 - 24*b*c*d**2*e**2 + 88*c**2*d
**3*e))/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)

________________________________________________________________________________________