3.553 \(\int \frac{x^5}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.201527, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2155, 697} \[ -\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}-\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}+\frac{x^3}{3 b c} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]

[Out]

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

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{x^5}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{a c+b c x+d \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{-a+x^2}{d+c x} \, dx,x,\sqrt{a+b x^3}\right )}{3 b^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d}{c^2}+\frac{x}{c}+\frac{-a c^2+d^2}{c^2 (d+c x)}\right ) \, dx,x,\sqrt{a+b x^3}\right )}{3 b^2}\\ &=\frac{x^3}{3 b c}-\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left (a c^2-d^2\right ) \log \left (d+c \sqrt{a+b x^3}\right )}{3 b^2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0692534, size = 63, normalized size = 0.86 \[ \frac{\left (2 d^2-2 a c^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )+c \left (b c x^3-2 d \sqrt{a+b x^3}\right )}{3 b^2 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]

[Out]

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

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Maple [C]  time = 0.015, size = 943, normalized size = 12.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)

[Out]

-2/3*d*(b*x^3+a)^(1/2)/c^2/b^2+1/3*I/b^4/d*2^(1/2)*sum((-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1
/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/
3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)
*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(
2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/
3))^(1/2),-1/2*c^2/b*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^2*b-I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*a*b-3*(-
a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/
3)))^(1/2)),_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2))*a-1/3*I*d/b^4/c^2*2^(1/2)*sum((-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/
b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3
)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3
))^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(
1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)
)*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),-1/2*c^2/b*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^2*b-I*(-a*b^2)^(2/3)*3^(1/2)*_
alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*
I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2))-1/3*a/c/b^2*ln(b*c^2*x^3+a*c^2-d^2)+1
/3*x^3/b/c+1/3/b^2/c^3*d^2*ln(b*c^2*x^3+a*c^2-d^2)

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Maxima [A]  time = 1.17846, size = 84, normalized size = 1.15 \begin{align*} \frac{\frac{{\left (b x^{3} + a\right )} c - 2 \, \sqrt{b x^{3} + a} d}{c^{2}} - \frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="maxima")

[Out]

1/3*(((b*x^3 + a)*c - 2*sqrt(b*x^3 + a)*d)/c^2 - 2*(a*c^2 - d^2)*log(sqrt(b*x^3 + a)*c + d)/c^3)/b^2

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Fricas [A]  time = 1.24101, size = 246, normalized size = 3.37 \begin{align*} \frac{b c^{2} x^{3} - 2 \, \sqrt{b x^{3} + a} c d -{\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) -{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right ) +{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="fricas")

[Out]

1/3*(b*c^2*x^3 - 2*sqrt(b*x^3 + a)*c*d - (a*c^2 - d^2)*log(b*c^2*x^3 + a*c^2 - d^2) - (a*c^2 - d^2)*log(sqrt(b
*x^3 + a)*c + d) + (a*c^2 - d^2)*log(sqrt(b*x^3 + a)*c - d))/(b^2*c^3)

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Sympy [A]  time = 4.27612, size = 95, normalized size = 1.3 \begin{align*} \begin{cases} \frac{2 \left (\frac{a + b x^{3}}{6 b c} - \frac{d \sqrt{a + b x^{3}}}{3 b c^{2}} - \frac{\left (a c^{2} - d^{2}\right ) \left (\begin{cases} \frac{\sqrt{a + b x^{3}}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (c \sqrt{a + b x^{3}} + d \right )}}{c} & \text{otherwise} \end{cases}\right )}{3 b c^{2}}\right )}{b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{2 \left (3 \sqrt{a} d + 3 a c\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

Piecewise((2*((a + b*x**3)/(6*b*c) - d*sqrt(a + b*x**3)/(3*b*c**2) - (a*c**2 - d**2)*Piecewise((sqrt(a + b*x**
3)/d, Eq(c, 0)), (log(c*sqrt(a + b*x**3) + d)/c, True))/(3*b*c**2))/b, Ne(b, 0)), (x**6/(2*(3*sqrt(a)*d + 3*a*
c)), True))

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Giac [A]  time = 1.11376, size = 97, normalized size = 1.33 \begin{align*} -\frac{\frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac{{\left (b x^{3} + a\right )} b c - 2 \, \sqrt{b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="giac")

[Out]

-1/3*(2*(a*c^2 - d^2)*log(abs(sqrt(b*x^3 + a)*c + d))/(b*c^3) - ((b*x^3 + a)*b*c - 2*sqrt(b*x^3 + a)*b*d)/(b^2
*c^2))/b