\(\int \frac {(b x+c x^2)^2}{(d+e x)^3} \, dx\) [238]

Optimal result

Integrand size = 19, antiderivative size = 119 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {c (3 c d-2 b e) x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac {2 d (c d-b e) (2 c d-b e)}{e^5 (d+e x)}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (d+e x)}{e^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)}{e^5}-\frac {d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac {2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac {c x (3 c d-2 b e)}{e^4}+\frac {c^2 x^2}{2 e^3} \]

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^3,x]

[Out]

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

Rule 712

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {-2 c e (3 c d-2 b e) x+c^2 e^2 x^2-\frac {d^2 (c d-b e)^2}{(d+e x)^2}+\frac {4 d \left (2 c^2 d^2-3 b c d e+b^2 e^2\right )}{d+e x}+2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (d+e x)}{2 e^5} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08

method	result	size
default	\(\frac {c \left (\frac {1}{2} c e \,x^{2}+2 b e x -3 c d x \right )}{e^{4}}+\frac {2 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}{2 e^{5} \left (e x +d \right )^{2}}+\frac {\left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(128\)
norman	\(\frac {\frac {c^{2} x^{4}}{2 e}+\frac {d^{2} \left (3 b^{2} e^{2}-18 b c d e +18 c^{2} d^{2}\right )}{2 e^{5}}+\frac {2 c \left (b e -c d \right ) x^{3}}{e^{2}}+\frac {2 d \left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) x}{e^{4}}}{\left (e x +d \right )^{2}}+\frac {\left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(131\)
risch	\(\frac {c^{2} x^{2}}{2 e^{3}}+\frac {2 c b x}{e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\left (2 b^{2} d \,e^{2}-6 b c e \,d^{2}+4 c^{2} d^{3}\right ) x +\frac {d^{2} \left (3 b^{2} e^{2}-10 b c d e +7 c^{2} d^{2}\right )}{2 e}}{e^{4} \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 \ln \left (e x +d \right ) b c d}{e^{4}}+\frac {6 \ln \left (e x +d \right ) c^{2} d^{2}}{e^{5}}\)	\(146\)
parallelrisch	\(\frac {c^{2} x^{4} e^{4}+2 \ln \left (e x +d \right ) x^{2} b^{2} e^{4}-12 \ln \left (e x +d \right ) x^{2} b c d \,e^{3}+12 \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{2}+4 x^{3} b c \,e^{4}-4 x^{3} c^{2} d \,e^{3}+4 \ln \left (e x +d \right ) x \,b^{2} d \,e^{3}-24 \ln \left (e x +d \right ) x b c \,d^{2} e^{2}+24 \ln \left (e x +d \right ) x \,c^{2} d^{3} e +2 \ln \left (e x +d \right ) b^{2} d^{2} e^{2}-12 \ln \left (e x +d \right ) b c \,d^{3} e +12 \ln \left (e x +d \right ) c^{2} d^{4}+4 x \,b^{2} d \,e^{3}-24 x b c \,d^{2} e^{2}+24 x \,c^{2} d^{3} e +3 b^{2} d^{2} e^{2}-18 d^{3} e b c +18 c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}\)	\(252\)

[In]

int((c*x^2+b*x)^2/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (115) = 230\).

Time = 0.27 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.00 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {c^{2} e^{4} x^{4} + 7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} - 4 \, {\left (c^{2} d e^{3} - b c e^{4}\right )} x^{3} - {\left (11 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3}\right )} x^{2} + 2 \, {\left (c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x + 2 \, {\left (6 \, c^{2} d^{4} - 6 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 2 \, {\left (6 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]

[In]

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

[Out]

1/2*(c^2*e^4*x^4 + 7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e^2 - 4*(c^2*d*e^3 - b*c*e^4)*x^3 - (11*c^2*d^2*e^2 -
8*b*c*d*e^3)*x^2 + 2*(c^2*d^3*e - 4*b*c*d^2*e^2 + 2*b^2*d*e^3)*x + 2*(6*c^2*d^4 - 6*b*c*d^3*e + b^2*d^2*e^2 +
(6*c^2*d^2*e^2 - 6*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(6*c^2*d^3*e - 6*b*c*d^2*e^2 + b^2*d*e^3)*x)*log(e*x + d))/(e^
7*x^2 + 2*d*e^6*x + d^2*e^5)

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.30 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {c^{2} x^{2}}{2 e^{3}} + x \left (\frac {2 b c}{e^{3}} - \frac {3 c^{2} d}{e^{4}}\right ) + \frac {3 b^{2} d^{2} e^{2} - 10 b c d^{3} e + 7 c^{2} d^{4} + x \left (4 b^{2} d e^{3} - 12 b c d^{2} e^{2} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} + \frac {\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} \]

[In]

integrate((c*x**2+b*x)**2/(e*x+d)**3,x)

[Out]

c**2*x**2/(2*e**3) + x*(2*b*c/e**3 - 3*c**2*d/e**4) + (3*b**2*d**2*e**2 - 10*b*c*d**3*e + 7*c**2*d**4 + x*(4*b
**2*d*e**3 - 12*b*c*d**2*e**2 + 8*c**2*d**3*e))/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + (b**2*e**2 - 6*b*c*
d*e + 6*c**2*d**2)*log(d + e*x)/e**5

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.24 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac {c^{2} e x^{2} - 2 \, {\left (3 \, c^{2} d - 2 \, b c e\right )} x}{2 \, e^{4}} + \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]

[In]

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

[Out]

1/2*(7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e^2 + 4*(2*c^2*d^3*e - 3*b*c*d^2*e^2 + b^2*d*e^3)*x)/(e^7*x^2 + 2*d*
e^6*x + d^2*e^5) + 1/2*(c^2*e*x^2 - 2*(3*c^2*d - 2*b*c*e)*x)/e^4 + (6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*log(e*x +
d)/e^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.17 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} + \frac {c^{2} e^{3} x^{2} - 6 \, c^{2} d e^{2} x + 4 \, b c e^{3} x}{2 \, e^{6}} + \frac {7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{5}} \]

[In]

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

[Out]

(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*log(abs(e*x + d))/e^5 + 1/2*(c^2*e^3*x^2 - 6*c^2*d*e^2*x + 4*b*c*e^3*x)/e^6
+ 1/2*(7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e^2 + 4*(2*c^2*d^3*e - 3*b*c*d^2*e^2 + b^2*d*e^3)*x)/((e*x + d)^2*
e^5)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.27 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {\frac {3\,b^2\,d^2\,e^2-10\,b\,c\,d^3\,e+7\,c^2\,d^4}{2\,e}+x\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )}{d^2\,e^4+2\,d\,e^5\,x+e^6\,x^2}-x\,\left (\frac {3\,c^2\,d}{e^4}-\frac {2\,b\,c}{e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{e^5}+\frac {c^2\,x^2}{2\,e^3} \]

[In]

int((b*x + c*x^2)^2/(d + e*x)^3,x)

[Out]

((7*c^2*d^4 + 3*b^2*d^2*e^2 - 10*b*c*d^3*e)/(2*e) + x*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e))/(d^2*e^4 + e^6*
x^2 + 2*d*e^5*x) - x*((3*c^2*d)/e^4 - (2*b*c)/e^3) + (log(d + e*x)*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e))/e^5 + (c
^2*x^2)/(2*e^3)