3.4.62 \(\int \frac {(d+e x)^4 (f+g x)^2}{(d^2-e^2 x^2)^2} \, dx\)

Optimal. Leaf size=107 \[ \frac {4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac {x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac {g x^2 (2 d g+e f)}{e}+\frac {g^2 x^3}{3} \]

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Rubi [A]  time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {848, 88} \begin {gather*} \frac {x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac {4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac {g x^2 (2 d g+e f)}{e}+\frac {g^2 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

((e^2*f^2 + 8*d*e*f*g + 8*d^2*g^2)*x)/e^2 + (g*(e*f + 2*d*g)*x^2)/e + (g^2*x^3)/3 + (4*d^2*(e*f + d*g)^2)/(e^3
*(d - e*x)) + (4*d*(e*f + d*g)*(e*f + 3*d*g)*Log[d - e*x])/e^3

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rubi steps

\begin {align*} \int \frac {(d+e x)^4 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^2 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {e^2 f^2+8 d e f g+8 d^2 g^2}{e^2}+\frac {2 g (e f+2 d g) x}{e}+g^2 x^2+\frac {4 d (-e f-3 d g) (e f+d g)}{e^2 (d-e x)}+\frac {4 d^2 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac {\left (e^2 f^2+8 d e f g+8 d^2 g^2\right ) x}{e^2}+\frac {g (e f+2 d g) x^2}{e}+\frac {g^2 x^3}{3}+\frac {4 d^2 (e f+d g)^2}{e^3 (d-e x)}+\frac {4 d (e f+d g) (e f+3 d g) \log (d-e x)}{e^3}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 115, normalized size = 1.07 \begin {gather*} -\frac {4 d^2 (d g+e f)^2}{e^3 (e x-d)}+\frac {x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d \left (3 d^2 g^2+4 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {g x^2 (2 d g+e f)}{e}+\frac {g^2 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

((e^2*f^2 + 8*d*e*f*g + 8*d^2*g^2)*x)/e^2 + (g*(e*f + 2*d*g)*x^2)/e + (g^2*x^3)/3 - (4*d^2*(e*f + d*g)^2)/(e^3
*(-d + e*x)) + (4*d*(e^2*f^2 + 4*d*e*f*g + 3*d^2*g^2)*Log[d - e*x])/e^3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2)^2, x]

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fricas [A]  time = 0.39, size = 206, normalized size = 1.93 \begin {gather*} \frac {e^{4} g^{2} x^{4} - 12 \, d^{2} e^{2} f^{2} - 24 \, d^{3} e f g - 12 \, d^{4} g^{2} + {\left (3 \, e^{4} f g + 5 \, d e^{3} g^{2}\right )} x^{3} + 3 \, {\left (e^{4} f^{2} + 7 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{2} - 3 \, {\left (d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 8 \, d^{3} e g^{2}\right )} x - 12 \, {\left (d^{2} e^{2} f^{2} + 4 \, d^{3} e f g + 3 \, d^{4} g^{2} - {\left (d e^{3} f^{2} + 4 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{3 \, {\left (e^{4} x - d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="fricas")

[Out]

1/3*(e^4*g^2*x^4 - 12*d^2*e^2*f^2 - 24*d^3*e*f*g - 12*d^4*g^2 + (3*e^4*f*g + 5*d*e^3*g^2)*x^3 + 3*(e^4*f^2 + 7
*d*e^3*f*g + 6*d^2*e^2*g^2)*x^2 - 3*(d*e^3*f^2 + 8*d^2*e^2*f*g + 8*d^3*e*g^2)*x - 12*(d^2*e^2*f^2 + 4*d^3*e*f*
g + 3*d^4*g^2 - (d*e^3*f^2 + 4*d^2*e^2*f*g + 3*d^3*e*g^2)*x)*log(e*x - d))/(e^4*x - d*e^3)

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giac [B]  time = 0.17, size = 250, normalized size = 2.34 \begin {gather*} 2 \, {\left (3 \, d^{3} g^{2} e^{3} + 4 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {1}{3} \, {\left (g^{2} x^{3} e^{12} + 6 \, d g^{2} x^{2} e^{11} + 24 \, d^{2} g^{2} x e^{10} + 3 \, f g x^{2} e^{12} + 24 \, d f g x e^{11} + 3 \, f^{2} x e^{12}\right )} e^{\left (-12\right )} + \frac {2 \, {\left (3 \, d^{4} g^{2} e^{4} + 4 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {4 \, {\left (d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} + d^{3} f^{2} e^{5} + {\left (d^{4} g^{2} e^{4} + 2 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{x^{2} e^{2} - d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="giac")

[Out]

2*(3*d^3*g^2*e^3 + 4*d^2*f*g*e^4 + d*f^2*e^5)*e^(-6)*log(abs(x^2*e^2 - d^2)) + 1/3*(g^2*x^3*e^12 + 6*d*g^2*x^2
*e^11 + 24*d^2*g^2*x*e^10 + 3*f*g*x^2*e^12 + 24*d*f*g*x*e^11 + 3*f^2*x*e^12)*e^(-12) + 2*(3*d^4*g^2*e^4 + 4*d^
3*f*g*e^5 + d^2*f^2*e^6)*e^(-7)*log(abs(2*x*e^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d) - 4*(d^5*g^2*e
^3 + 2*d^4*f*g*e^4 + d^3*f^2*e^5 + (d^4*g^2*e^4 + 2*d^3*f*g*e^5 + d^2*f^2*e^6)*x)*e^(-6)/(x^2*e^2 - d^2)

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maple [A]  time = 0.01, size = 167, normalized size = 1.56 \begin {gather*} \frac {g^{2} x^{3}}{3}+\frac {2 d \,g^{2} x^{2}}{e}+f g \,x^{2}-\frac {4 d^{4} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {8 d^{3} f g}{\left (e x -d \right ) e^{2}}+\frac {12 d^{3} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {4 d^{2} f^{2}}{\left (e x -d \right ) e}+\frac {16 d^{2} f g \ln \left (e x -d \right )}{e^{2}}+\frac {8 d^{2} g^{2} x}{e^{2}}+\frac {4 d \,f^{2} \ln \left (e x -d \right )}{e}+\frac {8 d f g x}{e}+f^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2)^2,x)

[Out]

1/3*g^2*x^3+2*d/e*g^2*x^2+f*g*x^2+8*d^2/e^2*g^2*x+8*d/e*f*g*x+f^2*x+12*d^3/e^3*g^2*ln(e*x-d)+16*d^2/e^2*f*g*ln
(e*x-d)+4*d/e*f^2*ln(e*x-d)-4*d^4/e^3/(e*x-d)*g^2-8*d^3/e^2/(e*x-d)*f*g-4*d^2/e/(e*x-d)*f^2

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maxima [A]  time = 0.45, size = 141, normalized size = 1.32 \begin {gather*} -\frac {4 \, {\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac {e^{2} g^{2} x^{3} + 3 \, {\left (e^{2} f g + 2 \, d e g^{2}\right )} x^{2} + 3 \, {\left (e^{2} f^{2} + 8 \, d e f g + 8 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} + \frac {4 \, {\left (d e^{2} f^{2} + 4 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="maxima")

[Out]

-4*(d^2*e^2*f^2 + 2*d^3*e*f*g + d^4*g^2)/(e^4*x - d*e^3) + 1/3*(e^2*g^2*x^3 + 3*(e^2*f*g + 2*d*e*g^2)*x^2 + 3*
(e^2*f^2 + 8*d*e*f*g + 8*d^2*g^2)*x)/e^2 + 4*(d*e^2*f^2 + 4*d^2*e*f*g + 3*d^3*g^2)*log(e*x - d)/e^3

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mupad [B]  time = 0.07, size = 185, normalized size = 1.73 \begin {gather*} x^2\,\left (\frac {g\,\left (d\,g+e\,f\right )}{e}+\frac {d\,g^2}{e}\right )+x\,\left (\frac {d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2}{e^2}+\frac {2\,d\,\left (\frac {2\,g\,\left (d\,g+e\,f\right )}{e}+\frac {2\,d\,g^2}{e}\right )}{e}-\frac {d^2\,g^2}{e^2}\right )+\frac {g^2\,x^3}{3}+\frac {4\,\left (d^4\,g^2+2\,d^3\,e\,f\,g+d^2\,e^2\,f^2\right )}{e\,\left (d\,e^2-e^3\,x\right )}+\frac {\ln \left (e\,x-d\right )\,\left (12\,d^3\,g^2+16\,d^2\,e\,f\,g+4\,d\,e^2\,f^2\right )}{e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^2*(d + e*x)^4)/(d^2 - e^2*x^2)^2,x)

[Out]

x^2*((g*(d*g + e*f))/e + (d*g^2)/e) + x*((d^2*g^2 + e^2*f^2 + 4*d*e*f*g)/e^2 + (2*d*((2*g*(d*g + e*f))/e + (2*
d*g^2)/e))/e - (d^2*g^2)/e^2) + (g^2*x^3)/3 + (4*(d^4*g^2 + d^2*e^2*f^2 + 2*d^3*e*f*g))/(e*(d*e^2 - e^3*x)) +
(log(e*x - d)*(12*d^3*g^2 + 4*d*e^2*f^2 + 16*d^2*e*f*g))/e^3

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sympy [A]  time = 0.74, size = 119, normalized size = 1.11 \begin {gather*} \frac {4 d \left (d g + e f\right ) \left (3 d g + e f\right ) \log {\left (- d + e x \right )}}{e^{3}} + \frac {g^{2} x^{3}}{3} + x^{2} \left (\frac {2 d g^{2}}{e} + f g\right ) + x \left (\frac {8 d^{2} g^{2}}{e^{2}} + \frac {8 d f g}{e} + f^{2}\right ) + \frac {- 4 d^{4} g^{2} - 8 d^{3} e f g - 4 d^{2} e^{2} f^{2}}{- d e^{3} + e^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)

[Out]

4*d*(d*g + e*f)*(3*d*g + e*f)*log(-d + e*x)/e**3 + g**2*x**3/3 + x**2*(2*d*g**2/e + f*g) + x*(8*d**2*g**2/e**2
+ 8*d*f*g/e + f**2) + (-4*d**4*g**2 - 8*d**3*e*f*g - 4*d**2*e**2*f**2)/(-d*e**3 + e**4*x)

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