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

________________________________________________________________________________________

Rubi [A]  time = 0.047441, 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, Rules used = {801, 635, 205, 260} \[ \frac{a^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}-\frac{a d \log \left (a+c x^2\right )}{2 c^2}-\frac{a e x}{c^2}+\frac{d x^2}{2 c}+\frac{e x^3}{3 c} \]

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x^3 (d+e x)}{a+c x^2} \, dx &=\int \left (-\frac{a e}{c^2}+\frac{d x}{c}+\frac{e x^2}{c}+\frac{a^2 e-a c d x}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac{a e x}{c^2}+\frac{d x^2}{2 c}+\frac{e x^3}{3 c}+\frac{\int \frac{a^2 e-a c d x}{a+c x^2} \, dx}{c^2}\\ &=-\frac{a e x}{c^2}+\frac{d x^2}{2 c}+\frac{e x^3}{3 c}-\frac{(a d) \int \frac{x}{a+c x^2} \, dx}{c}+\frac{\left (a^2 e\right ) \int \frac{1}{a+c x^2} \, dx}{c^2}\\ &=-\frac{a e x}{c^2}+\frac{d x^2}{2 c}+\frac{e x^3}{3 c}+\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}\\ \end{align*}

Mathematica [A]  time = 0.0398692, 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)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 65, normalized size = 0.9 \begin{align*}{\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}}}} \end{align*}

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(x*c/(a*c)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.65079, size = 331, normalized size = 4.53 \begin{align*} \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{c x \sqrt{\frac{a}{c}}}{a}\right ) - 6 \, a e x - 3 \, a d \log \left (c x^{2} + a\right )}{6 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(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(c*x*sqrt(a/c)/a) - 6*a*e*x - 3*a*d
*log(c*x^2 + a))/c^2]

________________________________________________________________________________________

Sympy [B]  time = 0.575053, size = 167, normalized size = 2.29 \begin{align*} - \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} \end{align*}

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*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)) + d*x**2/(2*c) + e*x**3/(3*c)

________________________________________________________________________________________

Giac [A]  time = 1.13927, size = 96, normalized size = 1.32 \begin{align*} \frac{a^{2} \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{\sqrt{a c} c^{2}} - \frac{a d \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(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*log(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