3.494 \(\int \frac{(d+e x)^5}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=190 \[ \frac{d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}+\frac{e^3 \left (5 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^3}-\frac{3 d e^2 x \left (2 c d^2-5 a e^2\right )}{2 a c^2}-\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(d+e x)^4 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

(-3*d*e^2*(2*c*d^2 - 5*a*e^2)*x)/(2*a*c^2) - (e^3*(2*c*d^2 - a*e^2)*x^2)/(a*c^2)
- (d*e^4*x^3)/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^4)/(2*a*c*(a + c*x^2)) + (d*(c
^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^
(5/2)) + (e^3*(5*c*d^2 - a*e^2)*Log[a + c*x^2])/c^3

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Rubi [A]  time = 0.423558, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}+\frac{e^3 \left (5 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^3}-\frac{3 d e^2 x \left (2 c d^2-5 a e^2\right )}{2 a c^2}-\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(d+e x)^4 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^5/(a + c*x^2)^2,x]

[Out]

(-3*d*e^2*(2*c*d^2 - 5*a*e^2)*x)/(2*a*c^2) - (e^3*(2*c*d^2 - a*e^2)*x^2)/(a*c^2)
- (d*e^4*x^3)/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^4)/(2*a*c*(a + c*x^2)) + (d*(c
^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^
(5/2)) + (e^3*(5*c*d^2 - a*e^2)*Log[a + c*x^2])/c^3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right ) \log{\left (a + c x^{2} \right )}}{c^{3}} - \frac{d e^{4} x^{3}}{2 a c} - \frac{\left (d + e x\right )^{4} \left (a e - c d x\right )}{2 a c \left (a + c x^{2}\right )} + \frac{2 e^{3} \left (a e^{2} - 2 c d^{2}\right ) \int x\, dx}{a c^{2}} + \frac{3 e^{2} \left (5 a e^{2} - 2 c d^{2}\right ) \int d\, dx}{2 a c^{2}} - \frac{d \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(c*x**2+a)**2,x)

[Out]

-e**3*(a*e**2 - 5*c*d**2)*log(a + c*x**2)/c**3 - d*e**4*x**3/(2*a*c) - (d + e*x)
**4*(a*e - c*d*x)/(2*a*c*(a + c*x**2)) + 2*e**3*(a*e**2 - 2*c*d**2)*Integral(x,
x)/(a*c**2) + 3*e**2*(5*a*e**2 - 2*c*d**2)*Integral(d, x)/(2*a*c**2) - d*(15*a**
2*e**4 - 10*a*c*d**2*e**2 - c**2*d**4)*atan(sqrt(c)*x/sqrt(a))/(2*a**(3/2)*c**(5
/2))

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Mathematica [A]  time = 0.2413, size = 164, normalized size = 0.86 \[ \frac{\frac{\sqrt{c} d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}+\frac{-a^3 e^5+5 a^2 c d e^3 (2 d+e x)-5 a c^2 d^3 e (d+2 e x)+c^3 d^5 x}{a \left (a+c x^2\right )}+2 \left (5 c d^2 e^3-a e^5\right ) \log \left (a+c x^2\right )+10 c d e^4 x+c e^5 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^5/(a + c*x^2)^2,x]

[Out]

(10*c*d*e^4*x + c*e^5*x^2 + (-(a^3*e^5) + c^3*d^5*x + 5*a^2*c*d*e^3*(2*d + e*x)
- 5*a*c^2*d^3*e*(d + 2*e*x))/(a*(a + c*x^2)) + (Sqrt[c]*d*(c^2*d^4 + 10*a*c*d^2*
e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*(5*c*d^2*e^3 - a*e^5)
*Log[a + c*x^2])/(2*c^3)

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Maple [A]  time = 0.016, size = 252, normalized size = 1.3 \[{\frac{{e}^{5}{x}^{2}}{2\,{c}^{2}}}+5\,{\frac{d{e}^{4}x}{{c}^{2}}}+{\frac{5\,adx{e}^{4}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-5\,{\frac{{d}^{3}x{e}^{2}}{c \left ( c{x}^{2}+a \right ) }}+{\frac{{d}^{5}x}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}-{\frac{{a}^{2}{e}^{5}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}+5\,{\frac{{e}^{3}a{d}^{2}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{5\,e{d}^{4}}{2\,c \left ( c{x}^{2}+a \right ) }}-{\frac{a\ln \left ( a \left ( c{x}^{2}+a \right ) \right ){e}^{5}}{{c}^{3}}}+5\,{\frac{\ln \left ( a \left ( c{x}^{2}+a \right ) \right ){d}^{2}{e}^{3}}{{c}^{2}}}-{\frac{15\,ad{e}^{4}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+5\,{\frac{{d}^{3}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{d}^{5}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(c*x^2+a)^2,x)

[Out]

1/2*e^5*x^2/c^2+5*e^4/c^2*d*x+5/2/c^2/(c*x^2+a)*d*a*x*e^4-5/c/(c*x^2+a)*d^3*x*e^
2+1/2/(c*x^2+a)*d^5/a*x-1/2/c^3/(c*x^2+a)*e^5*a^2+5/c^2/(c*x^2+a)*e^3*a*d^2-5/2/
c/(c*x^2+a)*e*d^4-1/c^3*a*ln(a*(c*x^2+a))*e^5+5/c^2*ln(a*(c*x^2+a))*d^2*e^3-15/2
/c^2*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d*e^4+5/c/(a*c)^(1/2)*arctan(c*x/(a*c
)^(1/2))*d^3*e^2+1/2/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228704, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} - 15 \, a^{3} c d e^{4} +{\left (c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} - 15 \, a^{2} c^{2} d e^{4}\right )} x^{2}\right )} \log \left (-\frac{2 \, a c x -{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (a c^{2} e^{5} x^{4} + 10 \, a c^{2} d e^{4} x^{3} + a^{2} c e^{5} x^{2} - 5 \, a c^{2} d^{4} e + 10 \, a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} x + 2 \,{\left (5 \, a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (5 \, a c^{2} d^{2} e^{3} - a^{2} c e^{5}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{4 \,{\left (a c^{4} x^{2} + a^{2} c^{3}\right )} \sqrt{-a c}}, \frac{{\left (a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} - 15 \, a^{3} c d e^{4} +{\left (c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} - 15 \, a^{2} c^{2} d e^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (a c^{2} e^{5} x^{4} + 10 \, a c^{2} d e^{4} x^{3} + a^{2} c e^{5} x^{2} - 5 \, a c^{2} d^{4} e + 10 \, a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} x + 2 \,{\left (5 \, a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (5 \, a c^{2} d^{2} e^{3} - a^{2} c e^{5}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \,{\left (a c^{4} x^{2} + a^{2} c^{3}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/4*((a*c^3*d^5 + 10*a^2*c^2*d^3*e^2 - 15*a^3*c*d*e^4 + (c^4*d^5 + 10*a*c^3*d^
3*e^2 - 15*a^2*c^2*d*e^4)*x^2)*log(-(2*a*c*x - (c*x^2 - a)*sqrt(-a*c))/(c*x^2 +
a)) - 2*(a*c^2*e^5*x^4 + 10*a*c^2*d*e^4*x^3 + a^2*c*e^5*x^2 - 5*a*c^2*d^4*e + 10
*a^2*c*d^2*e^3 - a^3*e^5 + (c^3*d^5 - 10*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*x + 2*(
5*a^2*c*d^2*e^3 - a^3*e^5 + (5*a*c^2*d^2*e^3 - a^2*c*e^5)*x^2)*log(c*x^2 + a))*s
qrt(-a*c))/((a*c^4*x^2 + a^2*c^3)*sqrt(-a*c)), 1/2*((a*c^3*d^5 + 10*a^2*c^2*d^3*
e^2 - 15*a^3*c*d*e^4 + (c^4*d^5 + 10*a*c^3*d^3*e^2 - 15*a^2*c^2*d*e^4)*x^2)*arct
an(sqrt(a*c)*x/a) + (a*c^2*e^5*x^4 + 10*a*c^2*d*e^4*x^3 + a^2*c*e^5*x^2 - 5*a*c^
2*d^4*e + 10*a^2*c*d^2*e^3 - a^3*e^5 + (c^3*d^5 - 10*a*c^2*d^3*e^2 + 15*a^2*c*d*
e^4)*x + 2*(5*a^2*c*d^2*e^3 - a^3*e^5 + (5*a*c^2*d^2*e^3 - a^2*c*e^5)*x^2)*log(c
*x^2 + a))*sqrt(a*c))/((a*c^4*x^2 + a^2*c^3)*sqrt(a*c))]

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Sympy [A]  time = 8.80712, size = 515, normalized size = 2.71 \[ \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} - \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{- 4 a^{3} e^{5} - 4 a^{2} c^{3} \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} - \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) + 20 a^{2} c d^{2} e^{3}}{15 a^{2} c d e^{4} - 10 a c^{2} d^{3} e^{2} - c^{3} d^{5}} \right )} + \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} + \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{- 4 a^{3} e^{5} - 4 a^{2} c^{3} \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} + \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) + 20 a^{2} c d^{2} e^{3}}{15 a^{2} c d e^{4} - 10 a c^{2} d^{3} e^{2} - c^{3} d^{5}} \right )} + \frac{- a^{3} e^{5} + 10 a^{2} c d^{2} e^{3} - 5 a c^{2} d^{4} e + x \left (5 a^{2} c d e^{4} - 10 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} + \frac{5 d e^{4} x}{c^{2}} + \frac{e^{5} x^{2}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(c*x**2+a)**2,x)

[Out]

(-e**3*(a*e**2 - 5*c*d**2)/c**3 - d*sqrt(-a**3*c**7)*(15*a**2*e**4 - 10*a*c*d**2
*e**2 - c**2*d**4)/(4*a**3*c**6))*log(x + (-4*a**3*e**5 - 4*a**2*c**3*(-e**3*(a*
e**2 - 5*c*d**2)/c**3 - d*sqrt(-a**3*c**7)*(15*a**2*e**4 - 10*a*c*d**2*e**2 - c*
*2*d**4)/(4*a**3*c**6)) + 20*a**2*c*d**2*e**3)/(15*a**2*c*d*e**4 - 10*a*c**2*d**
3*e**2 - c**3*d**5)) + (-e**3*(a*e**2 - 5*c*d**2)/c**3 + d*sqrt(-a**3*c**7)*(15*
a**2*e**4 - 10*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**6))*log(x + (-4*a**3*e**5 -
4*a**2*c**3*(-e**3*(a*e**2 - 5*c*d**2)/c**3 + d*sqrt(-a**3*c**7)*(15*a**2*e**4
- 10*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**6)) + 20*a**2*c*d**2*e**3)/(15*a**2*c
*d*e**4 - 10*a*c**2*d**3*e**2 - c**3*d**5)) + (-a**3*e**5 + 10*a**2*c*d**2*e**3
- 5*a*c**2*d**4*e + x*(5*a**2*c*d*e**4 - 10*a*c**2*d**3*e**2 + c**3*d**5))/(2*a*
*2*c**3 + 2*a*c**4*x**2) + 5*d*e**4*x/c**2 + e**5*x**2/(2*c**2)

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GIAC/XCAS [A]  time = 0.214082, size = 236, normalized size = 1.24 \[ \frac{{\left (5 \, c d^{2} e^{3} - a e^{5}\right )}{\rm ln}\left (c x^{2} + a\right )}{c^{3}} + \frac{{\left (c^{2} d^{5} + 10 \, a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} + \frac{c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4}}{2 \, c^{4}} - \frac{5 \, a c^{2} d^{4} e - 10 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} -{\left (c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*x^2 + a)^2,x, algorithm="giac")

[Out]

(5*c*d^2*e^3 - a*e^5)*ln(c*x^2 + a)/c^3 + 1/2*(c^2*d^5 + 10*a*c*d^3*e^2 - 15*a^2
*d*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2) + 1/2*(c^2*x^2*e^5 + 10*c^2*d*x*
e^4)/c^4 - 1/2*(5*a*c^2*d^4*e - 10*a^2*c*d^2*e^3 + a^3*e^5 - (c^3*d^5 - 10*a*c^2
*d^3*e^2 + 5*a^2*c*d*e^4)*x)/((c*x^2 + a)*a*c^3)