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

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

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Rubi [C]  time = 0.07, 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} \begin {gather*} \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}} \end {gather*}

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 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])

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])

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*}

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Mathematica [C]  time = 0.06, size = 86, normalized size = 0.50 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [C]  time = 2.56, size = 355, normalized size = 2.06 \begin {gather*} -\frac {i \left (\sqrt {3} \sqrt [3]{c}-i \sqrt [3]{c}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 (b c-a d)^{4/3}}-\frac {x}{\sqrt [3]{a+b x^3} (b c-a d)}+\frac {\left (\sqrt [3]{c}+i \sqrt {3} \sqrt [3]{c}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 (b c-a d)^{4/3}}-\frac {\sqrt {-1+i \sqrt {3}} \sqrt [3]{c} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{\sqrt {6} (b c-a d)^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/((b*c - a*d)*(a + b*x^3)^(1/3))) - (Sqrt[-1 + I*Sqrt[3]]*c^(1/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(
b*c - a*d)^(1/3)*x - (3*I)*c^(1/3)*(a + b*x^3)^(1/3) - Sqrt[3]*c^(1/3)*(a + b*x^3)^(1/3))])/(Sqrt[6]*(b*c - a*
d)^(4/3)) + ((c^(1/3) + I*Sqrt[3]*c^(1/3))*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/
3)])/(6*(b*c - a*d)^(4/3)) - ((I/12)*((-I)*c^(1/3) + Sqrt[3]*c^(1/3))*Log[(-2*I)*(b*c - a*d)^(2/3)*x^2 + c^(1/
3)*(b*c - a*d)^(1/3)*(I*x - Sqrt[3]*x)*(a + b*x^3)^(1/3) + (I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(b*c - a*
d)^(4/3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}

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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