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

Optimal. Leaf size=172 \[ -\frac{x}{\sqrt [3]{a+b x^3} (b c-a d)}+\frac{\sqrt [3]{c} \log \left (c+d x^3\right )}{6 (b c-a d)^{4/3}}-\frac{\sqrt [3]{c} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 (b c-a d)^{4/3}}+\frac{\sqrt [3]{c} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} (b c-a d)^{4/3}} \]

[Out]

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

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Rubi [C]  time = 0.0654884, antiderivative size = 92, normalized size of antiderivative = 0.53, 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} \[ \frac{x^4 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};-\frac{c \left (\frac{b x^3}{a}-\frac{d x^3}{c}\right )}{d x^3+c}\right )}{4 a c \sqrt [3]{a+b x^3} \left (\frac{d x^3}{c}+1\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^4*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, -((c*((b*x^3)/a - (d*x^3)/c))/(c + d*x^3))])/(4*a*
c*(a + b*x^3)^(1/3)*(1 + (d*x^3)/c)^(4/3))

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{x^3}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{x^3}{\left (1+\frac{b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=\frac{x^4 \sqrt [3]{1+\frac{b x^3}{a}} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};-\frac{c \left (\frac{b x^3}{a}-\frac{d x^3}{c}\right )}{c+d x^3}\right )}{4 a c \sqrt [3]{a+b x^3} \left (1+\frac{d x^3}{c}\right )^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.034121, size = 86, normalized size = 0.5 \[ \frac{x^4 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{4 a c \sqrt [3]{a+b x^3} \left (\frac{d x^3}{c}+1\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^4*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(4*a*c*(a +
b*x^3)^(1/3)*(1 + (d*x^3)/c)^(4/3))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/((a + b*x**3)**(4/3)*(c + d*x**3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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