3.705 \(\int \frac{(a+b x^3)^{4/3}}{x^5 (c+d x^3)} \, dx\)

Optimal. Leaf size=201 \[ -\frac{\sqrt [3]{a+b x^3} (5 b c-4 a d)}{4 c^2 x}+\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{7/3}}-\frac{(b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{7/3}}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{7/3}}-\frac{a \sqrt [3]{a+b x^3}}{4 c x^4} \]

[Out]

-(a*(a + b*x^3)^(1/3))/(4*c*x^4) - ((5*b*c - 4*a*d)*(a + b*x^3)^(1/3))/(4*c^2*x) - ((b*c - a*d)^(4/3)*ArcTan[(
1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(7/3)) + ((b*c - a*d)^(4/3)*Log[
c + d*x^3])/(6*c^(7/3)) - ((b*c - a*d)^(4/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(7/3
))

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Rubi [C]  time = 0.0635623, antiderivative size = 90, normalized size of antiderivative = 0.45, 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} \[ -\frac{a \sqrt [3]{a+b x^3} \left (\frac{d x^3}{c}+1\right )^{4/3} \, _2F_1\left (-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};-\frac{c \left (\frac{b x^3}{a}-\frac{d x^3}{c}\right )}{d x^3+c}\right )}{4 c x^4 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(4/3)/(x^5*(c + d*x^3)),x]

[Out]

-(a*(a + b*x^3)^(1/3)*(1 + (d*x^3)/c)^(4/3)*Hypergeometric2F1[-4/3, -4/3, -1/3, -((c*((b*x^3)/a - (d*x^3)/c))/
(c + d*x^3))])/(4*c*x^4*(1 + (b*x^3)/a)^(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 )^{4/3}}{x^5 \left (c+d x^3\right )} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{4/3}}{x^5 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{a \sqrt [3]{a+b x^3} \left (1+\frac{d x^3}{c}\right )^{4/3} \, _2F_1\left (-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};-\frac{c \left (\frac{b x^3}{a}-\frac{d x^3}{c}\right )}{c+d x^3}\right )}{4 c x^4 \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}

Mathematica [C]  time = 0.0328093, size = 84, normalized size = 0.42 \[ -\frac{a \sqrt [3]{a+b x^3} \left (\frac{d x^3}{c}+1\right )^{4/3} \, _2F_1\left (-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{4 c x^4 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(4/3)/(x^5*(c + d*x^3)),x]

[Out]

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

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(4/3)/x^5/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(4/3)/x^5/(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{4}{3}}}{{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(4/3)/x^5/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^5), 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)^(4/3)/x^5/(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{4}{3}}}{x^{5} \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)**(4/3)/x**5/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(4/3)/(x**5*(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{4}{3}}}{{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(4/3)/x^5/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^5), x)