3.688 \(\int \frac{(a+b x^3)^{2/3}}{x^6 (c+d x^3)} \, dx\)

Optimal. Leaf size=206 \[ -\frac{\left (a+b x^3\right )^{2/3} (2 b c-5 a d)}{10 a c^2 x^2}-\frac{d (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{8/3}}+\frac{d (b c-a d)^{2/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{8/3}}-\frac{d (b c-a d)^{2/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{8/3}}-\frac{\left (a+b x^3\right )^{2/3}}{5 c x^5} \]

[Out]

-(a + b*x^3)^(2/3)/(5*c*x^5) - ((2*b*c - 5*a*d)*(a + b*x^3)^(2/3))/(10*a*c^2*x^2) - (d*(b*c - a*d)^(2/3)*ArcTa
n[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(8/3)) - (d*(b*c - a*d)^(2/3)
*Log[c + d*x^3])/(6*c^(8/3)) + (d*(b*c - a*d)^(2/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2
*c^(8/3))

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Rubi [C]  time = 0.134778, antiderivative size = 148, normalized size of antiderivative = 0.72, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{-2 x^3 \left (2 c-3 d x^3\right ) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 x^3 \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+c \left (a+b x^3\right ) \left (2 c-3 d x^3\right )}{10 c^3 x^5 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^6*(c + d*x^3)),x]

[Out]

-(c*(a + b*x^3)*(2*c - 3*d*x^3) - 2*(b*c - a*d)*x^3*(2*c - 3*d*x^3)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d
)*x^3)/(c*(a + b*x^3))] + 6*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))])/(10*c^3*x^5*(a + b*x^3)^(1/3))

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^6 \left (c+d x^3\right )} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{\left (1+\frac{b x^3}{a}\right )^{2/3}}{x^6 \left (c+d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{c \left (a+b x^3\right ) \left (2 c-3 d x^3\right )-2 (b c-a d) x^3 \left (2 c-3 d x^3\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+6 (b c-a d) x^3 \left (c+d x^3\right ) \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{10 c^3 x^5 \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 0.389426, size = 148, normalized size = 0.72 \[ -\frac{2 x^3 \left (3 d x^3-2 c\right ) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 x^3 \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+c \left (a+b x^3\right ) \left (2 c-3 d x^3\right )}{10 c^3 x^5 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^6*(c + d*x^3)),x]

[Out]

-(c*(a + b*x^3)*(2*c - 3*d*x^3) + 2*(b*c - a*d)*x^3*(-2*c + 3*d*x^3)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*
d)*x^3)/(c*(a + b*x^3))] + 6*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(
a + b*x^3))])/(10*c^3*x^5*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^6/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(2/3)/x^6/(d*x^3+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^6/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^6), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^6/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{x^{6} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**6/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(2/3)/(x**6*(c + d*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^6/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^6), x)