3.6.50 \(\int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx\) [550]

Optimal. Leaf size=65 \[ -\frac {2 d g (e f+d g) x}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g}-\frac {2 d (e f+d g)^2 \log (d-e x)}{e^3} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 65, 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, 78} \begin {gather*} -\frac {2 d (d g+e f)^2 \log (d-e x)}{e^3}-\frac {2 d g x (d g+e f)}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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

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Mathematica [A]
time = 0.02, size = 73, normalized size = 1.12 \begin {gather*} -\frac {e x \left (6 d^2 g^2+3 d e g (4 f+g x)+e^2 \left (3 f^2+3 f g x+g^2 x^2\right )\right )+6 d (e f+d g)^2 \log (d-e x)}{3 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e*x*(6*d^2*g^2 + 3*d*e*g*(4*f + g*x) + e^2*(3*f^2 + 3*f*g*x + g^2*x^2)) + 6*d*(e*f + d*g)^2*Log[d - e*x]
)/e^3

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Maple [A]
time = 0.07, size = 95, normalized size = 1.46

method	result	size
norman	\(-\frac {g^{2} x^{3}}{3}-\frac {\left (2 d^{2} g^{2}+4 d e f g +e^{2} f^{2}\right ) x}{e^{2}}-\frac {g \left (d g +e f \right ) x^{2}}{e}-\frac {2 d \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(88\)
default	\(-\frac {\frac {1}{3} g^{2} x^{3} e^{2}+d e \,g^{2} x^{2}+e^{2} f g \,x^{2}+2 d^{2} g^{2} x +4 d e f g x +e^{2} f^{2} x}{e^{2}}-\frac {2 d \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(95\)
risch	\(-\frac {g^{2} x^{3}}{3}-\frac {d \,g^{2} x^{2}}{e}-f g \,x^{2}-\frac {2 d^{2} g^{2} x}{e^{2}}-\frac {4 d f g x}{e}-f^{2} x -\frac {2 d^{3} \ln \left (-e x +d \right ) g^{2}}{e^{3}}-\frac {4 d^{2} \ln \left (-e x +d \right ) f g}{e^{2}}-\frac {2 d \ln \left (-e x +d \right ) f^{2}}{e}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(g*x+f)^2/(-e^2*x^2+d^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 95, normalized size = 1.46 \begin {gather*} -2 \, {\left (d^{3} g^{2} + 2 \, 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 (d g^{2} e + f g e^{2}\right )} x^{2} + 3 \, {\left (2 \, d^{2} g^{2} + 4 \, d f g e + f^{2} e^{2}\right )} x\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d^3*g^2 + 2*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*(d*g^2*e + f*g*e^2)*x^2 + 3*
(2*d^2*g^2 + 4*d*f*g*e + f^2*e^2)*x)*e^(-2)

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Fricas [A]
time = 2.56, size = 94, normalized size = 1.45 \begin {gather*} -\frac {1}{3} \, {\left (6 \, d^{2} g^{2} x e + {\left (g^{2} x^{3} + 3 \, f g x^{2} + 3 \, f^{2} x\right )} e^{3} + 3 \, {\left (d g^{2} x^{2} + 4 \, d f g x\right )} e^{2} + 6 \, {\left (d^{3} g^{2} + 2 \, d^{2} f g e + d f^{2} e^{2}\right )} \log \left (x e - d\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.17, size = 70, normalized size = 1.08 \begin {gather*} - \frac {2 d \left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} - \frac {g^{2} x^{3}}{3} - x^{2} \left (\frac {d g^{2}}{e} + f g\right ) - x \left (\frac {2 d^{2} g^{2}}{e^{2}} + \frac {4 d f g}{e} + f^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.80, size = 100, normalized size = 1.54 \begin {gather*} -2 \, {\left (d^{3} g^{2} + 2 \, 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^{3} + 3 \, d g^{2} x^{2} e^{2} + 6 \, d^{2} g^{2} x e + 3 \, f g x^{2} e^{3} + 12 \, d f g x e^{2} + 3 \, f^{2} x e^{3}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d^3*g^2 + 2*d^2*f*g*e + d*f^2*e^2)*e^(-3)*log(abs(x*e - d)) - 1/3*(g^2*x^3*e^3 + 3*d*g^2*x^2*e^2 + 6*d^2*g
^2*x*e + 3*f*g*x^2*e^3 + 12*d*f*g*x*e^2 + 3*f^2*x*e^3)*e^(-3)

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Mupad [B]
time = 0.07, size = 127, normalized size = 1.95 \begin {gather*} -x^2\,\left (\frac {d\,g^2+2\,e\,f\,g}{2\,e}+\frac {d\,g^2}{2\,e}\right )-x\,\left (\frac {e\,f^2+2\,d\,g\,f}{e}+\frac {d\,\left (\frac {d\,g^2+2\,e\,f\,g}{e}+\frac {d\,g^2}{e}\right )}{e}\right )-\frac {g^2\,x^3}{3}-\frac {\ln \left (e\,x-d\right )\,\left (2\,d^3\,g^2+4\,d^2\,e\,f\,g+2\,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)^2)/(d^2 - e^2*x^2),x)

[Out]

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

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