3.554 \(\int \frac {x^2}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx\)

Optimal. Leaf size=26 \[ \frac {2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b c} \]

[Out]

2/3*ln(d+c*(b*x^3+a)^(1/2))/b/c

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Rubi [A]  time = 0.11, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2155, 31} \[ \frac {2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 26, normalized size = 1.00 \[ \frac {2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 61, normalized size = 2.35 \[ \frac {\log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + \log \left (\sqrt {b x^{3} + a} c + d\right ) - \log \left (\sqrt {b x^{3} + a} c - d\right )}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.32, size = 23, normalized size = 0.88 \[ \frac {2 \, \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 455, normalized size = 17.50 \[ \frac {\ln \left (b \,c^{2} x^{3}+a \,c^{2}-d^{2}\right )}{3 b c}-\frac {i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )^{2} b^{2}+i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right ) b -\left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right ) b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {\left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )^{2} b +i \sqrt {3}\, a b -3 a b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )\right ) c^{2}}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{3 b^{3} d \sqrt {b \,x^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 22, normalized size = 0.85 \[ \frac {2 \, \log \left (\sqrt {b x^{3} + a} c + d\right )}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*log(sqrt(b*x^3 + a)*c + d)/(b*c)

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mupad [B]  time = 3.51, size = 60, normalized size = 2.31 \[ \frac {\ln \left (\frac {d+c\,\sqrt {b\,x^3+a}}{d-c\,\sqrt {b\,x^3+a}}\right )+\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )}{3\,b\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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