3.6.48 \(\int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx\) [548]

Optimal. Leaf size=141 \[ -\frac {d^2 \left (7 e^2 f^2+16 d e f g+8 d^2 g^2\right ) x}{e^2}-\frac {d \left (2 e^2 f^2+7 d e f g+4 d^2 g^2\right ) x^2}{e}-\frac {1}{3} (e f+d g) (e f+7 d g) x^3-\frac {1}{2} e g (e f+2 d g) x^4-\frac {1}{5} e^2 g^2 x^5-\frac {8 d^3 (e f+d g)^2 \log (d-e x)}{e^3} \]

[Out]

-d^2*(8*d^2*g^2+16*d*e*f*g+7*e^2*f^2)*x/e^2-d*(4*d^2*g^2+7*d*e*f*g+2*e^2*f^2)*x^2/e-1/3*(d*g+e*f)*(7*d*g+e*f)*
x^3-1/2*e*g*(2*d*g+e*f)*x^4-1/5*e^2*g^2*x^5-8*d^3*(d*g+e*f)^2*ln(-e*x+d)/e^3

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Rubi [A]
time = 0.12, antiderivative size = 141, 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 {8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac {d x^2 \left (4 d^2 g^2+7 d e f g+2 e^2 f^2\right )}{e}-\frac {d^2 x \left (8 d^2 g^2+16 d e f g+7 e^2 f^2\right )}{e^2}-\frac {1}{2} e g x^4 (2 d g+e f)-\frac {1}{3} x^3 (d g+e f) (7 d g+e f)-\frac {1}{5} e^2 g^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*(7*e^2*f^2 + 16*d*e*f*g + 8*d^2*g^2)*x)/e^2) - (d*(2*e^2*f^2 + 7*d*e*f*g + 4*d^2*g^2)*x^2)/e - ((e*f +
d*g)*(e*f + 7*d*g)*x^3)/3 - (e*g*(e*f + 2*d*g)*x^4)/2 - (e^2*g^2*x^5)/5 - (8*d^3*(e*f + d*g)^2*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}{d^2-e^2 x^2} \, dx &=\int \frac {(d+e x)^3 (f+g x)^2}{d-e x} \, dx\\ &=\int \left (-\frac {d^2 \left (7 e^2 f^2+16 d e f g+8 d^2 g^2\right )}{e^2}-\frac {2 d \left (2 e^2 f^2+7 d e f g+4 d^2 g^2\right ) x}{e}+(-e f-7 d g) (e f+d g) x^2-2 e g (e f+2 d g) x^3-e^2 g^2 x^4-\frac {8 d^3 (e f+d g)^2}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac {d^2 \left (7 e^2 f^2+16 d e f g+8 d^2 g^2\right ) x}{e^2}-\frac {d \left (2 e^2 f^2+7 d e f g+4 d^2 g^2\right ) x^2}{e}-\frac {1}{3} (e f+d g) (e f+7 d g) x^3-\frac {1}{2} e g (e f+2 d g) x^4-\frac {1}{5} e^2 g^2 x^5-\frac {8 d^3 (e f+d g)^2 \log (d-e x)}{e^3}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 134, normalized size = 0.95 \begin {gather*} -\frac {x \left (240 d^4 g^2+120 d^3 e g (4 f+g x)+70 d^2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+10 d e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )+e^4 x^2 \left (10 f^2+15 f g x+6 g^2 x^2\right )\right )}{30 e^2}-\frac {8 d^3 (e f+d g)^2 \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),x]

[Out]

-1/30*(x*(240*d^4*g^2 + 120*d^3*e*g*(4*f + g*x) + 70*d^2*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) + 10*d*e^3*x*(6*f^2 +
8*f*g*x + 3*g^2*x^2) + e^4*x^2*(10*f^2 + 15*f*g*x + 6*g^2*x^2)))/e^2 - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3

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Maple [A]
time = 0.08, size = 179, normalized size = 1.27

method	result	size
norman	\(\left (-\frac {7}{3} d^{2} g^{2}-\frac {8}{3} d e f g -\frac {1}{3} e^{2} f^{2}\right ) x^{3}-\frac {e^{2} g^{2} x^{5}}{5}-\frac {d \left (4 d^{2} g^{2}+7 d e f g +2 e^{2} f^{2}\right ) x^{2}}{e}-\frac {d^{2} \left (8 d^{2} g^{2}+16 d e f g +7 e^{2} f^{2}\right ) x}{e^{2}}-\frac {e g \left (2 d g +e f \right ) x^{4}}{2}-\frac {8 d^{3} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(155\)
default	\(-\frac {\frac {1}{5} g^{2} e^{4} x^{5}+d \,e^{3} g^{2} x^{4}+\frac {1}{2} e^{4} f g \,x^{4}+\frac {7}{3} d^{2} e^{2} g^{2} x^{3}+\frac {8}{3} d \,e^{3} f g \,x^{3}+\frac {1}{3} e^{4} f^{2} x^{3}+4 d^{3} e \,g^{2} x^{2}+7 d^{2} e^{2} f g \,x^{2}+2 d \,e^{3} f^{2} x^{2}+8 d^{4} g^{2} x +16 d^{3} e f g x +7 d^{2} e^{2} f^{2} x}{e^{2}}-\frac {8 d^{3} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(179\)
risch	\(-\frac {e^{2} g^{2} x^{5}}{5}-e d \,g^{2} x^{4}-\frac {e^{2} f g \,x^{4}}{2}-\frac {7 d^{2} g^{2} x^{3}}{3}-\frac {8 e d f g \,x^{3}}{3}-\frac {e^{2} f^{2} x^{3}}{3}-\frac {4 d^{3} g^{2} x^{2}}{e}-7 d^{2} f g \,x^{2}-2 e d \,f^{2} x^{2}-\frac {8 d^{4} g^{2} x}{e^{2}}-\frac {16 d^{3} f g x}{e}-7 d^{2} f^{2} x -\frac {8 d^{5} \ln \left (-e x +d \right ) g^{2}}{e^{3}}-\frac {16 d^{4} \ln \left (-e x +d \right ) f g}{e^{2}}-\frac {8 d^{3} \ln \left (-e x +d \right ) f^{2}}{e}\)	\(183\)

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),x,method=_RETURNVERBOSE)

[Out]

-1/e^2*(1/5*g^2*e^4*x^5+d*e^3*g^2*x^4+1/2*e^4*f*g*x^4+7/3*d^2*e^2*g^2*x^3+8/3*d*e^3*f*g*x^3+1/3*e^4*f^2*x^3+4*
d^3*e*g^2*x^2+7*d^2*e^2*f*g*x^2+2*d*e^3*f^2*x^2+8*d^4*g^2*x+16*d^3*e*f*g*x+7*d^2*e^2*f^2*x)-8*d^3*(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 = 0.29, size = 167, normalized size = 1.18 \begin {gather*} -8 \, {\left (d^{5} g^{2} + 2 \, d^{4} f g e + d^{3} f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right ) - \frac {1}{30} \, {\left (6 \, g^{2} x^{5} e^{4} + 15 \, {\left (2 \, d g^{2} e^{3} + f g e^{4}\right )} x^{4} + 10 \, {\left (7 \, d^{2} g^{2} e^{2} + 8 \, d f g e^{3} + f^{2} e^{4}\right )} x^{3} + 30 \, {\left (4 \, d^{3} g^{2} e + 7 \, d^{2} f g e^{2} + 2 \, d f^{2} e^{3}\right )} x^{2} + 30 \, {\left (8 \, d^{4} g^{2} + 16 \, d^{3} f g e + 7 \, d^{2} 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)^4*(g*x+f)^2/(-e^2*x^2+d^2),x, algorithm="maxima")

[Out]

-8*(d^5*g^2 + 2*d^4*f*g*e + d^3*f^2*e^2)*e^(-3)*log(x*e - d) - 1/30*(6*g^2*x^5*e^4 + 15*(2*d*g^2*e^3 + f*g*e^4
)*x^4 + 10*(7*d^2*g^2*e^2 + 8*d*f*g*e^3 + f^2*e^4)*x^3 + 30*(4*d^3*g^2*e + 7*d^2*f*g*e^2 + 2*d*f^2*e^3)*x^2 +
30*(8*d^4*g^2 + 16*d^3*f*g*e + 7*d^2*f^2*e^2)*x)*e^(-2)

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Fricas [A]
time = 3.78, size = 168, normalized size = 1.19 \begin {gather*} -\frac {1}{30} \, {\left (240 \, d^{4} g^{2} x e + {\left (6 \, g^{2} x^{5} + 15 \, f g x^{4} + 10 \, f^{2} x^{3}\right )} e^{5} + 10 \, {\left (3 \, d g^{2} x^{4} + 8 \, d f g x^{3} + 6 \, d f^{2} x^{2}\right )} e^{4} + 70 \, {\left (d^{2} g^{2} x^{3} + 3 \, d^{2} f g x^{2} + 3 \, d^{2} f^{2} x\right )} e^{3} + 120 \, {\left (d^{3} g^{2} x^{2} + 4 \, d^{3} f g x\right )} e^{2} + 240 \, {\left (d^{5} g^{2} + 2 \, d^{4} f g e + d^{3} 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)^4*(g*x+f)^2/(-e^2*x^2+d^2),x, algorithm="fricas")

[Out]

-1/30*(240*d^4*g^2*x*e + (6*g^2*x^5 + 15*f*g*x^4 + 10*f^2*x^3)*e^5 + 10*(3*d*g^2*x^4 + 8*d*f*g*x^3 + 6*d*f^2*x
^2)*e^4 + 70*(d^2*g^2*x^3 + 3*d^2*f*g*x^2 + 3*d^2*f^2*x)*e^3 + 120*(d^3*g^2*x^2 + 4*d^3*f*g*x)*e^2 + 240*(d^5*
g^2 + 2*d^4*f*g*e + d^3*f^2*e^2)*log(x*e - d))*e^(-3)

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Sympy [A]
time = 0.27, size = 150, normalized size = 1.06 \begin {gather*} - \frac {8 d^{3} \left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} - \frac {e^{2} g^{2} x^{5}}{5} - x^{4} \left (d e g^{2} + \frac {e^{2} f g}{2}\right ) - x^{3} \cdot \left (\frac {7 d^{2} g^{2}}{3} + \frac {8 d e f g}{3} + \frac {e^{2} f^{2}}{3}\right ) - x^{2} \cdot \left (\frac {4 d^{3} g^{2}}{e} + 7 d^{2} f g + 2 d e f^{2}\right ) - x \left (\frac {8 d^{4} g^{2}}{e^{2}} + \frac {16 d^{3} f g}{e} + 7 d^{2} f^{2}\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),x)

[Out]

-8*d**3*(d*g + e*f)**2*log(-d + e*x)/e**3 - e**2*g**2*x**5/5 - x**4*(d*e*g**2 + e**2*f*g/2) - x**3*(7*d**2*g**
2/3 + 8*d*e*f*g/3 + e**2*f**2/3) - x**2*(4*d**3*g**2/e + 7*d**2*f*g + 2*d*e*f**2) - x*(8*d**4*g**2/e**2 + 16*d
**3*f*g/e + 7*d**2*f**2)

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Giac [A]
time = 1.07, size = 177, normalized size = 1.26 \begin {gather*} -8 \, {\left (d^{5} g^{2} + 2 \, d^{4} f g e + d^{3} f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right ) - \frac {1}{30} \, {\left (6 \, g^{2} x^{5} e^{7} + 30 \, d g^{2} x^{4} e^{6} + 70 \, d^{2} g^{2} x^{3} e^{5} + 120 \, d^{3} g^{2} x^{2} e^{4} + 240 \, d^{4} g^{2} x e^{3} + 15 \, f g x^{4} e^{7} + 80 \, d f g x^{3} e^{6} + 210 \, d^{2} f g x^{2} e^{5} + 480 \, d^{3} f g x e^{4} + 10 \, f^{2} x^{3} e^{7} + 60 \, d f^{2} x^{2} e^{6} + 210 \, d^{2} f^{2} x e^{5}\right )} e^{\left (-5\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),x, algorithm="giac")

[Out]

-8*(d^5*g^2 + 2*d^4*f*g*e + d^3*f^2*e^2)*e^(-3)*log(abs(x*e - d)) - 1/30*(6*g^2*x^5*e^7 + 30*d*g^2*x^4*e^6 + 7
0*d^2*g^2*x^3*e^5 + 120*d^3*g^2*x^2*e^4 + 240*d^4*g^2*x*e^3 + 15*f*g*x^4*e^7 + 80*d*f*g*x^3*e^6 + 210*d^2*f*g*
x^2*e^5 + 480*d^3*f*g*x*e^4 + 10*f^2*x^3*e^7 + 60*d*f^2*x^2*e^6 + 210*d^2*f^2*x*e^5)*e^(-5)

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Mupad [B]
time = 0.11, size = 351, normalized size = 2.49 \begin {gather*} -x^2\,\left (\frac {d^3\,g^2+6\,d^2\,e\,f\,g+3\,d\,e^2\,f^2}{2\,e}+\frac {d\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{e}+\frac {d\,\left (e\,g\,\left (3\,d\,g+2\,e\,f\right )+d\,e\,g^2\right )}{e}\right )}{2\,e}\right )-x^3\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{3\,e}+\frac {d\,\left (e\,g\,\left (3\,d\,g+2\,e\,f\right )+d\,e\,g^2\right )}{3\,e}\right )-x^4\,\left (\frac {e\,g\,\left (3\,d\,g+2\,e\,f\right )}{4}+\frac {d\,e\,g^2}{4}\right )-x\,\left (\frac {d\,\left (\frac {d^3\,g^2+6\,d^2\,e\,f\,g+3\,d\,e^2\,f^2}{e}+\frac {d\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{e}+\frac {d\,\left (e\,g\,\left (3\,d\,g+2\,e\,f\right )+d\,e\,g^2\right )}{e}\right )}{e}\right )}{e}+\frac {d^2\,f\,\left (2\,d\,g+3\,e\,f\right )}{e}\right )-\frac {\ln \left (e\,x-d\right )\,\left (8\,d^5\,g^2+16\,d^4\,e\,f\,g+8\,d^3\,e^2\,f^2\right )}{e^3}-\frac {e^2\,g^2\,x^5}{5} \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),x)

[Out]

- x^2*((d^3*g^2 + 3*d*e^2*f^2 + 6*d^2*e*f*g)/(2*e) + (d*((e^3*f^2 + 3*d^2*e*g^2 + 6*d*e^2*f*g)/e + (d*(e*g*(3*
d*g + 2*e*f) + d*e*g^2))/e))/(2*e)) - x^3*((e^3*f^2 + 3*d^2*e*g^2 + 6*d*e^2*f*g)/(3*e) + (d*(e*g*(3*d*g + 2*e*
f) + d*e*g^2))/(3*e)) - x^4*((e*g*(3*d*g + 2*e*f))/4 + (d*e*g^2)/4) - x*((d*((d^3*g^2 + 3*d*e^2*f^2 + 6*d^2*e*
f*g)/e + (d*((e^3*f^2 + 3*d^2*e*g^2 + 6*d*e^2*f*g)/e + (d*(e*g*(3*d*g + 2*e*f) + d*e*g^2))/e))/e))/e + (d^2*f*
(2*d*g + 3*e*f))/e) - (log(e*x - d)*(8*d^5*g^2 + 8*d^3*e^2*f^2 + 16*d^4*e*f*g))/e^3 - (e^2*g^2*x^5)/5

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