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

Optimal. Leaf size=119 \[ \frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)}{e^5}-\frac {d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac {2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac {c x (3 c d-2 b e)}{e^4}+\frac {c^2 x^2}{2 e^3} \]

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Rubi [A]  time = 0.11, antiderivative size = 119, 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 {\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} \end {gather*}

Antiderivative was successfully verified.

[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 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)^3} \, dx &=\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*}

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Mathematica [A]  time = 0.07, size = 116, normalized size = 0.97 \begin {gather*} \frac {\frac {4 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )}{d+e x}+2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)-\frac {d^2 (c d-b e)^2}{(d+e x)^2}-2 c e x (3 c d-2 b e)+c^2 e^2 x^2}{2 e^5} \end {gather*}

Antiderivative was successfully verified.

[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)

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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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 238, normalized size = 2.00 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.17, size = 134, normalized size = 1.13 \begin {gather*} {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c^{2} x^{2} e^{3} - 6 \, c^{2} d x e^{2} + 4 \, b c x e^{3}\right )} e^{\left (-6\right )} + \frac {{\left (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\right )} e^{\left (-5\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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maple [A]  time = 0.08, size = 178, normalized size = 1.50 \begin {gather*} -\frac {b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {b c \,d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {c^{2} x^{2}}{2 e^{3}}+\frac {2 b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 b c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {6 b c d \ln \left (e x +d \right )}{e^{4}}+\frac {2 b c x}{e^{3}}+\frac {4 c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 c^{2} d x}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 147, normalized size = 1.24 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.08, size = 151, normalized size = 1.27 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.94, size = 155, normalized size = 1.30 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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