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*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)

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Rubi [A]  time = 0.0145199, antiderivative size = 21, normalized size of antiderivative = 1., 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 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]

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{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*}

Mathematica [A]  time = 0.0028395, size = 21, normalized size = 1. \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

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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Maple [B]  time = 0.005, size = 218, normalized size = 10.4 \begin{align*}{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{6\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C}{6}\sqrt [3]{a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \end{align*}

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

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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((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]

Exception raised: ValueError

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Fricas [A]  time = 1.01843, size = 51, normalized size = 2.43 \begin{align*} \frac{C \log \left (b x + a^{\frac{1}{3}} b^{\frac{2}{3}}\right )}{b^{\frac{1}{3}}} \end{align*}

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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Sympy [A]  time = 0.311516, size = 20, normalized size = 0.95 \begin{align*} \frac{C \log{\left (\sqrt [3]{a} b^{\frac{2}{3}} + b x \right )}}{\sqrt [3]{b}} \end{align*}

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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Giac [A]  time = 1.47152, size = 22, normalized size = 1.05 \begin{align*} \frac{C \log \left ({\left | b^{\frac{1}{3}} x + a^{\frac{1}{3}} \right |}\right )}{b^{\frac{1}{3}}} \end{align*}

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)