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

Optimal. Leaf size=73 \[ \frac{a^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}-\frac{a d \log \left (a+c x^2\right )}{2 c^2}-\frac{a e x}{c^2}+\frac{d x^2}{2 c}+\frac{e x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.112093, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{a^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}-\frac{a d \log \left (a+c x^2\right )}{2 c^2}-\frac{a e x}{c^2}+\frac{d x^2}{2 c}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*e*atan(sqrt(c)*x/sqrt(a))/c**(5/2) - a*d*log(a + c*x**2)/(2*c**2) + d*I
ntegral(x, x)/c + e*x**3/(3*c) - e*Integral(a, x)/c**2

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Mathematica [A]  time = 0.0709745, size = 64, normalized size = 0.88 \[ \frac{a^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}+\frac{x (c x (3 d+2 e x)-6 a e)-3 a d \log \left (a+c x^2\right )}{6 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^(3/2)*e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/c^(5/2) + (x*(-6*a*e + c*x*(3*d + 2*e*x)
) - 3*a*d*Log[a + c*x^2])/(6*c^2)

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Maple [A]  time = 0.005, size = 65, normalized size = 0.9 \[{\frac{e{x}^{3}}{3\,c}}+{\frac{d{x}^{2}}{2\,c}}-{\frac{aex}{{c}^{2}}}-{\frac{ad\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}+{\frac{{a}^{2}e}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x+d)/(c*x^2+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283216, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, c e x^{3} + 3 \, c d x^{2} + 3 \, a e \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) - 6 \, a e x - 3 \, a d \log \left (c x^{2} + a\right )}{6 \, c^{2}}, \frac{2 \, c e x^{3} + 3 \, c d x^{2} + 6 \, a e \sqrt{\frac{a}{c}} \arctan \left (\frac{x}{\sqrt{\frac{a}{c}}}\right ) - 6 \, a e x - 3 \, a d \log \left (c x^{2} + a\right )}{6 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.88517, size = 167, normalized size = 2.29 \[ - \frac{a e x}{c^{2}} + \left (- \frac{a d}{2 c^{2}} - \frac{e \sqrt{- a^{3} c^{5}}}{2 c^{5}}\right ) \log{\left (x + \frac{a d + 2 c^{2} \left (- \frac{a d}{2 c^{2}} - \frac{e \sqrt{- a^{3} c^{5}}}{2 c^{5}}\right )}{a e} \right )} + \left (- \frac{a d}{2 c^{2}} + \frac{e \sqrt{- a^{3} c^{5}}}{2 c^{5}}\right ) \log{\left (x + \frac{a d + 2 c^{2} \left (- \frac{a d}{2 c^{2}} + \frac{e \sqrt{- a^{3} c^{5}}}{2 c^{5}}\right )}{a e} \right )} + \frac{d x^{2}}{2 c} + \frac{e x^{3}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269811, size = 96, normalized size = 1.32 \[ \frac{a^{2} \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{\sqrt{a c} c^{2}} - \frac{a d{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 6 \, a c x e}{6 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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