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*Log[d + c*Sqrt[a + b*x^3]])/(3*b*c)

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Rubi [A]  time = 0.110529, antiderivative size = 26, normalized size of antiderivative = 1., 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 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 31

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

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*}

Mathematica [A]  time = 0.0283146, size = 26, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]

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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Maple [C]  time = 0.013, size = 455, normalized size = 17.5 \begin{align*}{\frac{-{\frac{i}{3}}\sqrt{2}}{{b}^{3}d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}b{c}^{2}+{c}^{2}a-{d}^{2} \right ) }{\sqrt [3]{-a{b}^{2}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}-i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( i\sqrt [3]{-a{b}^{2}}\sqrt{3}{\it \_alpha}\,b-i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+2\,{{\it \_alpha}}^{2}{b}^{2}-\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b- \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},-{\frac{{c}^{2}}{2\,b{d}^{2}} \left ( 2\,i\sqrt [3]{-a{b}^{2}}\sqrt{3}{{\it \_alpha}}^{2}b-i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}{\it \_alpha}+i\sqrt{3}ab-3\, \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}-3\,ab \right ) },\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}}}+{\frac{\ln \left ( b{c}^{2}{x}^{3}+{c}^{2}a-{d}^{2} \right ) }{3\,bc}} \end{align*}

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*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)*_al
pha*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/b/c*ln(b*c^2*x^3+a*c^2-d^2)

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Maxima [A]  time = 1.19344, size = 30, normalized size = 1.15 \begin{align*} \frac{2 \, \log \left (\sqrt{b x^{3} + a} c + d\right )}{3 \, b c} \end{align*}

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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Fricas [B]  time = 1.26852, size = 135, normalized size = 5.19 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 2.50954, size = 32, normalized size = 1.23 \begin{align*} \frac{2 \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} \end{align*}

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]

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

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Giac [A]  time = 1.12645, size = 31, normalized size = 1.19 \begin{align*} \frac{2 \, \log \left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \, b c} \end{align*}

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)