3.573 \(\int \frac {(d+e x)^3 (f+g x)^2}{(d^2-e^2 x^2)^3} \, dx\)

Optimal. Leaf size=61 \[ -\frac {2 g (d g+e f)}{e^3 (d-e x)}+\frac {(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac {g^2 \log (d-e x)}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 61, 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, 43} \[ -\frac {2 g (d g+e f)}{e^3 (d-e x)}+\frac {(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac {g^2 \log (d-e x)}{e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

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Mathematica [A]  time = 0.03, size = 49, normalized size = 0.80 \[ \frac {\frac {(d g+e f) (e (f+4 g x)-3 d g)}{(d-e x)^2}-2 g^2 \log (d-e x)}{2 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 100, normalized size = 1.64 \[ \frac {e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \, {\left (e^{2} f g + d e g^{2}\right )} x - 2 \, {\left (e^{2} g^{2} x^{2} - 2 \, d e g^{2} x + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.20, size = 195, normalized size = 3.20 \[ -\frac {d g^{2} e^{\left (-3\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} - \frac {1}{2} \, g^{2} e^{\left (-3\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {{\left (4 \, {\left (d^{2} g^{2} e^{4} + d f g e^{5}\right )} x^{3} + {\left (5 \, d^{3} g^{2} e^{3} + 6 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} x^{2} - 2 \, {\left (d^{4} g^{2} e^{2} - d^{2} f^{2} e^{4}\right )} x - {\left (3 \, d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} - d^{3} f^{2} e^{5}\right )} e^{\left (-2\right )}\right )} e^{\left (-4\right )}}{2 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 105, normalized size = 1.72 \[ \frac {d^{2} g^{2}}{2 \left (e x -d \right )^{2} e^{3}}+\frac {d f g}{\left (e x -d \right )^{2} e^{2}}+\frac {f^{2}}{2 \left (e x -d \right )^{2} e}+\frac {2 d \,g^{2}}{\left (e x -d \right ) e^{3}}+\frac {2 f g}{\left (e x -d \right ) e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(g*x+f)^2/(-e^2*x^2+d^2)^3,x)

[Out]

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

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maxima [A]  time = 0.44, size = 81, normalized size = 1.33 \[ \frac {e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \, {\left (e^{2} f g + d e g^{2}\right )} x}{2 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} - \frac {g^{2} \log \left (e x - d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.07, size = 80, normalized size = 1.31 \[ -\frac {\frac {3\,d^2\,g^2+2\,d\,e\,f\,g-e^2\,f^2}{2\,e^3}-\frac {2\,g\,x\,\left (d\,g+e\,f\right )}{e^2}}{d^2-2\,d\,e\,x+e^2\,x^2}-\frac {g^2\,\ln \left (e\,x-d\right )}{e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^2*(d + e*x)^3)/(d^2 - e^2*x^2)^3,x)

[Out]

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

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sympy [A]  time = 0.54, size = 83, normalized size = 1.36 \[ - \frac {3 d^{2} g^{2} + 2 d e f g - e^{2} f^{2} + x \left (- 4 d e g^{2} - 4 e^{2} f g\right )}{2 d^{2} e^{3} - 4 d e^{4} x + 2 e^{5} x^{2}} - \frac {g^{2} \log {\left (- d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(g*x+f)**2/(-e**2*x**2+d**2)**3,x)

[Out]

-(3*d**2*g**2 + 2*d*e*f*g - e**2*f**2 + x*(-4*d*e*g**2 - 4*e**2*f*g))/(2*d**2*e**3 - 4*d*e**4*x + 2*e**5*x**2)
- g**2*log(-d + e*x)/e**3

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