3.43 \(\int \frac {a^{2/3} C-\sqrt [3]{a} \sqrt [3]{b} C x+b^{2/3} C x^2}{a+b x^3} \, dx\)

Optimal. Leaf size=21 \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

[Out]

C*ln(a^(1/3)+b^(1/3)*x)/b^(1/3)

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1586, 31} \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[(a^(2/3)*C - a^(1/3)*b^(1/3)*C*x + b^(2/3)*C*x^2)/(a + b*x^3),x]

[Out]

(C*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)

Rule 31

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

\begin {align*} \int \frac {a^{2/3} C-\sqrt [3]{a} \sqrt [3]{b} C x+b^{2/3} C x^2}{a+b x^3} \, dx &=\int \frac {1}{\frac {\sqrt [3]{a}}{C}+\frac {\sqrt [3]{b} x}{C}} \, dx\\ &=\frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^(2/3)*C - a^(1/3)*b^(1/3)*C*x + b^(2/3)*C*x^2)/(a + b*x^3),x]

[Out]

(C*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)

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fricas [A]  time = 0.86, size = 17, normalized size = 0.81 \[ \frac {C \log \left (b x + a^{\frac {1}{3}} b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*log(b*x + a^(1/3)*b^(2/3))/b^(1/3)

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giac [A]  time = 0.31, size = 16, normalized size = 0.76 \[ \frac {C \log \left ({\left | b^{\frac {1}{3}} x + a^{\frac {1}{3}} \right |}\right )}{b^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*log(abs(b^(1/3)*x + a^(1/3)))/b^(1/3)

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maple [B]  time = 0.05, size = 218, normalized size = 10.38 \[ \frac {\sqrt {3}\, C \,a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {C \,a^{\frac {2}{3}} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {C \,a^{\frac {2}{3}} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {\sqrt {3}\, C \,a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{\frac {2}{3}}}+\frac {C \,a^{\frac {1}{3}} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{\frac {2}{3}}}-\frac {C \,a^{\frac {1}{3}} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{\frac {2}{3}}}+\frac {C \ln \left (b \,x^{3}+a \right )}{3 b^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(2/3)*C-a^(1/3)*b^(1/3)*C*x+b^(2/3)*C*x^2)/(b*x^3+a),x)

[Out]

1/3/(a/b)^(2/3)*C*a^(2/3)/b*ln(x+(a/b)^(1/3))-1/6/(a/b)^(2/3)*C*a^(2/3)/b*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/
3/(a/b)^(2/3)*3^(1/2)*C*a^(2/3)/b*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*C/b^(2/3)*a^(1/3)/(a/b)^(1/3)*ln
(x+(a/b)^(1/3))-1/6*C/b^(2/3)*a^(1/3)/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-1/3*C/b^(2/3)*a^(1/3)*3^(1
/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*C/b^(1/3)*ln(b*x^3+a)

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maxima [B]  time = 2.99, size = 210, normalized size = 10.00 \[ -\frac {\sqrt {3} {\left (2 \, C a b^{\frac {2}{3}} + {\left (3 \, C a^{\frac {1}{3}} b^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, C a^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, C a}{b^{\frac {1}{3}}}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (2 \, C b^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - C a^{\frac {1}{3}} b^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} - C a^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C b^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} + C a^{\frac {1}{3}} b^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + C a^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(3)*(2*C*a*b^(2/3) + (3*C*a^(1/3)*b^(1/3)*(a/b)^(2/3) - 3*C*a^(2/3)*(a/b)^(1/3) - 2*C*a/b^(1/3))*b)*a
rctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b) + 1/6*(2*C*b^(2/3)*(a/b)^(2/3) - C*a^(1/3)*b^(1/3)*(a
/b)^(1/3) - C*a^(2/3))*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b*(a/b)^(2/3)) + 1/3*(C*b^(2/3)*(a/b)^(2/3) + C
*a^(1/3)*b^(1/3)*(a/b)^(1/3) + C*a^(2/3))*log(x + (a/b)^(1/3))/(b*(a/b)^(2/3))

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mupad [B]  time = 4.90, size = 15, normalized size = 0.71 \[ \frac {C\,\ln \left (x+\frac {a^{1/3}}{b^{1/3}}\right )}{b^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*a^(2/3) + C*b^(2/3)*x^2 - C*a^(1/3)*b^(1/3)*x)/(a + b*x^3),x)

[Out]

(C*log(x + a^(1/3)/b^(1/3)))/b^(1/3)

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sympy [A]  time = 0.26, size = 20, normalized size = 0.95 \[ \frac {C \log {\left (\sqrt [3]{a} b^{\frac {2}{3}} + b x \right )}}{\sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**(2/3)*C-a**(1/3)*b**(1/3)*C*x+b**(2/3)*C*x**2)/(b*x**3+a),x)

[Out]

C*log(a**(1/3)*b**(2/3) + b*x)/b**(1/3)

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