3.488 \(\int \frac{(d+e x)^3}{a+c x^2} \, dx\)

Optimal. Leaf size=90 \[ \frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.182336, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a + c*x^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.101015, size = 80, normalized size = 0.89 \[ \frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (\left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )+c e x (6 d+e x)\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(a + c*x^2),x]

[Out]

(d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(c*e*x*
(6*d + e*x) + (3*c*d^2 - a*e^2)*Log[a + c*x^2]))/(2*c^2)

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Maple [A]  time = 0.007, size = 99, normalized size = 1.1 \[{\frac{{e}^{3}{x}^{2}}{2\,c}}+3\,{\frac{d{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ){e}^{3}a}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ){d}^{2}e}{2\,c}}-3\,{\frac{ad{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{{d}^{3}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+a),x)

[Out]

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

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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)^3/(c*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21811, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} \log \left (-\frac{2 \, a c x -{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) -{\left (c e^{3} x^{2} + 6 \, c d e^{2} x +{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{2 \, \sqrt{-a c} c^{2}}, \frac{2 \,{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (c e^{3} x^{2} + 6 \, c d e^{2} x +{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \, \sqrt{a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + a),x, algorithm="fricas")

[Out]

[-1/2*((c^2*d^3 - 3*a*c*d*e^2)*log(-(2*a*c*x - (c*x^2 - a)*sqrt(-a*c))/(c*x^2 +
a)) - (c*e^3*x^2 + 6*c*d*e^2*x + (3*c*d^2*e - a*e^3)*log(c*x^2 + a))*sqrt(-a*c))
/(sqrt(-a*c)*c^2), 1/2*(2*(c^2*d^3 - 3*a*c*d*e^2)*arctan(sqrt(a*c)*x/a) + (c*e^3
*x^2 + 6*c*d*e^2*x + (3*c*d^2*e - a*e^3)*log(c*x^2 + a))*sqrt(a*c))/(sqrt(a*c)*c
^2)]

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Sympy [A]  time = 3.37138, size = 308, normalized size = 3.42 \[ \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{- a^{2} e^{3} - 2 a c^{2} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{- a^{2} e^{3} - 2 a c^{2} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \frac{3 d e^{2} x}{c} + \frac{e^{3} x^{2}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*x**2+a),x)

[Out]

(-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4
))*log(x + (-a**2*e**3 - 2*a*c**2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c
**5)*(3*a*e**2 - c*d**2)/(2*a*c**4)) + 3*a*c*d**2*e)/(3*a*c*d*e**2 - c**2*d**3))
+ (-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c
**4))*log(x + (-a**2*e**3 - 2*a*c**2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-
a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4)) + 3*a*c*d**2*e)/(3*a*c*d*e**2 - c**2*d**
3)) + 3*d*e**2*x/c + e**3*x**2/(2*c)

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GIAC/XCAS [A]  time = 0.212045, size = 105, normalized size = 1.17 \[ \frac{{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} + \frac{{\left (3 \, c d^{2} e - a e^{3}\right )}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{c x^{2} e^{3} + 6 \, c d x e^{2}}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + a),x, algorithm="giac")

[Out]

(c*d^3 - 3*a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*(3*c*d^2*e - a*e^3
)*ln(c*x^2 + a)/c^2 + 1/2*(c*x^2*e^3 + 6*c*d*x*e^2)/c^2