3.3.37 \(\int \frac {(b x+c x^2)^2}{(d+e x)^2} \, dx\) [237]

Optimal. Leaf size=107 \[ \frac {(c d-b e) (3 c d-b e) x}{e^4}-\frac {c (c d-b e) x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac {2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 107, 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 = {712} \begin {gather*} -\frac {d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac {2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac {x (c d-b e) (3 c d-b e)}{e^4}-\frac {c x^2 (c d-b e)}{e^3}+\frac {c^2 x^3}{3 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^2,x]

[Out]

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

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Mathematica [A]
time = 0.07, size = 114, normalized size = 1.07 \begin {gather*} \frac {3 e \left (3 c^2 d^2-4 b c d e+b^2 e^2\right ) x-3 c e^2 (c d-b e) x^2+c^2 e^3 x^3-\frac {3 d^2 (c d-b e)^2}{d+e x}-6 d \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.45, size = 131, normalized size = 1.22

method	result	size
default	\(\frac {\frac {1}{3} c^{2} e^{2} x^{3}+b c \,e^{2} x^{2}-c^{2} d e \,x^{2}+b^{2} e^{2} x -4 b c d e x +3 d^{2} c^{2} x}{e^{4}}-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {2 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(131\)
norman	\(\frac {\frac {\left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) x^{2}}{e^{3}}+\frac {\left (2 d^{2} e^{2} b^{2}-6 d^{3} e b c +4 c^{2} d^{4}\right ) x}{d \,e^{4}}+\frac {c^{2} x^{4}}{3 e}+\frac {c \left (3 b e -2 c d \right ) x^{3}}{3 e^{2}}}{e x +d}-\frac {2 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(139\)
risch	\(\frac {c^{2} x^{3}}{3 e^{2}}+\frac {b c \,x^{2}}{e^{2}}-\frac {c^{2} d \,x^{2}}{e^{3}}+\frac {b^{2} x}{e^{2}}-\frac {4 b c d x}{e^{3}}+\frac {3 d^{2} c^{2} x}{e^{4}}-\frac {d^{2} b^{2}}{e^{3} \left (e x +d \right )}+\frac {2 d^{3} b c}{e^{4} \left (e x +d \right )}-\frac {d^{4} c^{2}}{e^{5} \left (e x +d \right )}-\frac {2 d \ln \left (e x +d \right ) b^{2}}{e^{3}}+\frac {6 d^{2} \ln \left (e x +d \right ) b c}{e^{4}}-\frac {4 d^{3} \ln \left (e x +d \right ) c^{2}}{e^{5}}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 134, normalized size = 1.25 \begin {gather*} -2 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {1}{3} \, {\left (c^{2} x^{3} e^{2} - 3 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2} + 3 \, {\left (3 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} x\right )} e^{\left (-4\right )} - \frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}}{x e^{6} + d 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.05, size = 197, normalized size = 1.84 \begin {gather*} -\frac {3 \, c^{2} d^{4} - {\left (c^{2} x^{4} + 3 \, b c x^{3} + 3 \, b^{2} x^{2}\right )} e^{4} + {\left (2 \, c^{2} d x^{3} + 9 \, b c d x^{2} - 3 \, b^{2} d x\right )} e^{3} - 3 \, {\left (2 \, c^{2} d^{2} x^{2} - 4 \, b c d^{2} x - b^{2} d^{2}\right )} e^{2} - 3 \, {\left (3 \, c^{2} d^{3} x + 2 \, b c d^{3}\right )} e + 6 \, {\left (2 \, c^{2} d^{4} + b^{2} d x e^{3} - {\left (3 \, b c d^{2} x - b^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{2} d^{3} x - 3 \, b c d^{3}\right )} e\right )} \log \left (x e + d\right )}{3 \, {\left (x e^{6} + d 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)^2,x, algorithm="fricas")

[Out]

-1/3*(3*c^2*d^4 - (c^2*x^4 + 3*b*c*x^3 + 3*b^2*x^2)*e^4 + (2*c^2*d*x^3 + 9*b*c*d*x^2 - 3*b^2*d*x)*e^3 - 3*(2*c
^2*d^2*x^2 - 4*b*c*d^2*x - b^2*d^2)*e^2 - 3*(3*c^2*d^3*x + 2*b*c*d^3)*e + 6*(2*c^2*d^4 + b^2*d*x*e^3 - (3*b*c*
d^2*x - b^2*d^2)*e^2 + (2*c^2*d^3*x - 3*b*c*d^3)*e)*log(x*e + d))/(x*e^6 + d*e^5)

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Sympy [A]
time = 0.27, size = 126, normalized size = 1.18 \begin {gather*} \frac {c^{2} x^{3}}{3 e^{2}} - \frac {2 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log {\left (d + e x \right )}}{e^{5}} + x^{2} \left (\frac {b c}{e^{2}} - \frac {c^{2} d}{e^{3}}\right ) + x \left (\frac {b^{2}}{e^{2}} - \frac {4 b c d}{e^{3}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {- b^{2} d^{2} e^{2} + 2 b c d^{3} e - c^{2} d^{4}}{d e^{5} + e^{6} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.89, size = 184, normalized size = 1.72 \begin {gather*} \frac {1}{3} \, {\left (c^{2} - \frac {3 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {3 \, {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-5\right )} + 2 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{2} d^{4} e^{3}}{x e + d} - \frac {2 \, b c d^{3} e^{4}}{x e + d} + \frac {b^{2} d^{2} e^{5}}{x e + d}\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.18, size = 158, normalized size = 1.48 \begin {gather*} x\,\left (\frac {b^2}{e^2}+\frac {2\,d\,\left (\frac {2\,c^2\,d}{e^3}-\frac {2\,b\,c}{e^2}\right )}{e}-\frac {c^2\,d^2}{e^4}\right )-x^2\,\left (\frac {c^2\,d}{e^3}-\frac {b\,c}{e^2}\right )-\frac {b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4}{e\,\left (x\,e^5+d\,e^4\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )}{e^5}+\frac {c^2\,x^3}{3\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2/(d + e*x)^2,x)

[Out]

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

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