3.6.60 \(\int \frac {(d+e x)^4 (f+g x)^2}{(d^2-e^2 x^2)^2} \, dx\) [560]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.09, 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 = {862, 90} \begin {gather*} \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} \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 90

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 862

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/e)*x)^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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 115, normalized size = 1.07 \begin {gather*} \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 \left (e^2 f^2+4 d e f g+3 d^2 g^2\right ) \log (d-e x)}{e^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

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 133, normalized size = 1.24

method	result	size
default	\(\frac {\frac {1}{3} g^{2} x^{3} e^{2}+2 d e \,g^{2} x^{2}+e^{2} f g \,x^{2}+8 d^{2} g^{2} x +8 d e f g x +e^{2} f^{2} x}{e^{2}}+\frac {4 d^{2} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right )}{e^{3} \left (-e x +d \right )}+\frac {4 d \left (3 d^{2} g^{2}+4 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(133\)
risch	\(\frac {g^{2} x^{3}}{3}+\frac {2 d \,g^{2} x^{2}}{e}+f g \,x^{2}+\frac {8 d^{2} g^{2} x}{e^{2}}+\frac {8 d f g x}{e}+f^{2} x +\frac {4 d^{4} g^{2}}{e^{3} \left (-e x +d \right )}+\frac {8 d^{3} f g}{e^{2} \left (-e x +d \right )}+\frac {4 d^{2} f^{2}}{e \left (-e x +d \right )}+\frac {12 d^{3} \ln \left (-e x +d \right ) g^{2}}{e^{3}}+\frac {16 d^{2} \ln \left (-e x +d \right ) f g}{e^{2}}+\frac {4 d \ln \left (-e x +d \right ) f^{2}}{e}\)	\(161\)
norman	\(\frac {\left (-\frac {23}{3} d^{2} g^{2}-8 d e f g -e^{2} f^{2}\right ) x^{3}+\frac {d^{2} \left (6 d^{3} g^{2}+9 d^{2} e f g +4 d \,e^{2} f^{2}\right )}{e^{3}}+\frac {d^{2} \left (12 d^{2} g^{2}+16 d e f g +5 e^{2} f^{2}\right ) x}{e^{2}}-\frac {e^{2} g^{2} x^{5}}{3}-e g \left (2 d g +e f \right ) x^{4}}{-e^{2} x^{2}+d^{2}}+\frac {4 d \left (3 d^{2} g^{2}+4 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(170\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 137, normalized size = 1.28 \begin {gather*} 4 \, {\left (3 \, d^{3} g^{2} + 4 \, d^{2} f g e + d f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right ) + \frac {1}{3} \, {\left (g^{2} x^{3} e^{2} + 3 \, {\left (2 \, d g^{2} e + f g e^{2}\right )} x^{2} + 3 \, {\left (8 \, d^{2} g^{2} + 8 \, d f g e + f^{2} e^{2}\right )} x\right )} e^{\left (-2\right )} - \frac {4 \, {\left (d^{4} g^{2} + 2 \, d^{3} f g e + d^{2} f^{2} e^{2}\right )}}{x e^{4} - d 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*(3*d^3*g^2 + 4*d^2*f*g*e + d*f^2*e^2)*e^(-3)*log(x*e - d) + 1/3*(g^2*x^3*e^2 + 3*(2*d*g^2*e + f*g*e^2)*x^2 +
3*(8*d^2*g^2 + 8*d*f*g*e + f^2*e^2)*x)*e^(-2) - 4*(d^4*g^2 + 2*d^3*f*g*e + d^2*f^2*e^2)/(x*e^4 - d*e^3)

________________________________________________________________________________________

Fricas [A]
time = 2.76, size = 201, normalized size = 1.88 \begin {gather*} -\frac {12 \, d^{4} g^{2} - {\left (g^{2} x^{4} + 3 \, f g x^{3} + 3 \, f^{2} x^{2}\right )} e^{4} - {\left (5 \, d g^{2} x^{3} + 21 \, d f g x^{2} - 3 \, d f^{2} x\right )} e^{3} - 6 \, {\left (3 \, d^{2} g^{2} x^{2} - 4 \, d^{2} f g x - 2 \, d^{2} f^{2}\right )} e^{2} + 24 \, {\left (d^{3} g^{2} x + d^{3} f g\right )} e + 12 \, {\left (3 \, d^{4} g^{2} - d f^{2} x e^{3} - {\left (4 \, d^{2} f g x - d^{2} f^{2}\right )} e^{2} - {\left (3 \, d^{3} g^{2} x - 4 \, d^{3} f g\right )} e\right )} \log \left (x e - d\right )}{3 \, {\left (x e^{4} - 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*(12*d^4*g^2 - (g^2*x^4 + 3*f*g*x^3 + 3*f^2*x^2)*e^4 - (5*d*g^2*x^3 + 21*d*f*g*x^2 - 3*d*f^2*x)*e^3 - 6*(3
*d^2*g^2*x^2 - 4*d^2*f*g*x - 2*d^2*f^2)*e^2 + 24*(d^3*g^2*x + d^3*f*g)*e + 12*(3*d^4*g^2 - d*f^2*x*e^3 - (4*d^
2*f*g*x - d^2*f^2)*e^2 - (3*d^3*g^2*x - 4*d^3*f*g)*e)*log(x*e - d))/(x*e^4 - d*e^3)

________________________________________________________________________________________

Sympy [A]
time = 0.33, 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} \cdot \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)

________________________________________________________________________________________

Giac [A]
time = 1.37, size = 141, normalized size = 1.32 \begin {gather*} 4 \, {\left (3 \, d^{3} g^{2} + 4 \, d^{2} f g e + d f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right ) + \frac {1}{3} \, {\left (g^{2} x^{3} e^{6} + 6 \, d g^{2} x^{2} e^{5} + 24 \, d^{2} g^{2} x e^{4} + 3 \, f g x^{2} e^{6} + 24 \, d f g x e^{5} + 3 \, f^{2} x e^{6}\right )} e^{\left (-6\right )} - \frac {4 \, {\left (d^{4} g^{2} + 2 \, d^{3} f g e + d^{2} f^{2} e^{2}\right )} e^{\left (-3\right )}}{x e - d} \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]

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

________________________________________________________________________________________

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

________________________________________________________________________________________