3.279 \(\int \frac{x^4 (d+e x)}{a+c x^2} \, dx\)

Optimal. Leaf size=87 \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}+\frac{a^2 e \log \left (a+c x^2\right )}{2 c^3}-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.0621743, antiderivative size = 87, 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} \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}+\frac{a^2 e \log \left (a+c x^2\right )}{2 c^3}-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x))/(a + c*x^2),x]

[Out]

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

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^4 (d+e x)}{a+c x^2} \, dx &=\int \left (-\frac{a d}{c^2}-\frac{a e x}{c^2}+\frac{d x^2}{c}+\frac{e x^3}{c}+\frac{a^2 d+a^2 e x}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c}+\frac{\int \frac{a^2 d+a^2 e x}{a+c x^2} \, dx}{c^2}\\ &=-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c}+\frac{\left (a^2 d\right ) \int \frac{1}{a+c x^2} \, dx}{c^2}+\frac{\left (a^2 e\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}\\ &=-\frac{a d x}{c^2}-\frac{a e x^2}{2 c^2}+\frac{d x^3}{3 c}+\frac{e x^4}{4 c}+\frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{5/2}}+\frac{a^2 e \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.035014, size = 75, normalized size = 0.86 \[ \frac{12 a^{3/2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+6 a^2 e \log \left (a+c x^2\right )+c x \left (c x^2 (4 d+3 e x)-6 a (2 d+e x)\right )}{12 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x))/(a + c*x^2),x]

[Out]

(c*x*(-6*a*(2*d + e*x) + c*x^2*(4*d + 3*e*x)) + 12*a^(3/2)*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]] + 6*a^2*e*Log
[a + c*x^2])/(12*c^3)

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Maple [A]  time = 0.006, size = 77, normalized size = 0.9 \begin{align*}{\frac{e{x}^{4}}{4\,c}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}-{\frac{adx}{{c}^{2}}}+{\frac{{a}^{2}e\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{3}}}+{\frac{{a}^{2}d}{{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^4*(e*x+d)/(c*x^2+a),x)

[Out]

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

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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^4*(e*x+d)/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48693, size = 402, normalized size = 4.62 \begin{align*} \left [\frac{3 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} - 6 \, a c e x^{2} + 6 \, a c d \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) - 12 \, a c d x + 6 \, a^{2} e \log \left (c x^{2} + a\right )}{12 \, c^{3}}, \frac{3 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} - 6 \, a c e x^{2} + 12 \, a c d \sqrt{\frac{a}{c}} \arctan \left (\frac{c x \sqrt{\frac{a}{c}}}{a}\right ) - 12 \, a c d x + 6 \, a^{2} e \log \left (c x^{2} + a\right )}{12 \, c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+a),x, algorithm="fricas")

[Out]

[1/12*(3*c^2*e*x^4 + 4*c^2*d*x^3 - 6*a*c*e*x^2 + 6*a*c*d*sqrt(-a/c)*log((c*x^2 + 2*c*x*sqrt(-a/c) - a)/(c*x^2
+ a)) - 12*a*c*d*x + 6*a^2*e*log(c*x^2 + a))/c^3, 1/12*(3*c^2*e*x^4 + 4*c^2*d*x^3 - 6*a*c*e*x^2 + 12*a*c*d*sqr
t(a/c)*arctan(c*x*sqrt(a/c)/a) - 12*a*c*d*x + 6*a^2*e*log(c*x^2 + a))/c^3]

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Sympy [B]  time = 0.604295, size = 189, normalized size = 2.17 \begin{align*} - \frac{a d x}{c^{2}} - \frac{a e x^{2}}{2 c^{2}} + \left (\frac{a^{2} e}{2 c^{3}} - \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right ) \log{\left (x + \frac{- a^{2} e + 2 c^{3} \left (\frac{a^{2} e}{2 c^{3}} - \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right )}{a c d} \right )} + \left (\frac{a^{2} e}{2 c^{3}} + \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right ) \log{\left (x + \frac{- a^{2} e + 2 c^{3} \left (\frac{a^{2} e}{2 c^{3}} + \frac{d \sqrt{- a^{3} c^{7}}}{2 c^{6}}\right )}{a c d} \right )} + \frac{d x^{3}}{3 c} + \frac{e x^{4}}{4 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(c*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.14675, size = 115, normalized size = 1.32 \begin{align*} \frac{a^{2} d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{a^{2} e \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 6 \, a c^{2} x^{2} e - 12 \, a c^{2} d x}{12 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+a),x, algorithm="giac")

[Out]

a^2*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c^2) + 1/2*a^2*e*log(c*x^2 + a)/c^3 + 1/12*(3*c^3*x^4*e + 4*c^3*d*x^3 -
6*a*c^2*x^2*e - 12*a*c^2*d*x)/c^4