3.4.67 \(\int \frac {x^3 (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\)

Optimal. Leaf size=229 \[ \frac {a \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{4/3} d}-\frac {a \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{4/3} d}+\frac {5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 b^{4/3} d}-\frac {5 a \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3} d}+\frac {2^{2/3} a \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d}-\frac {x \left (a+b x^3\right )^{2/3}}{3 b d} \]

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Rubi [C]  time = 0.06, antiderivative size = 66, normalized size of antiderivative = 0.29, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {511, 510} \begin {gather*} \frac {x^4 \left (a+b x^3\right )^{2/3} F_1\left (\frac {4}{3};-\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{4 a d \left (\frac {b x^3}{a}+1\right )^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^4*(a + b*x^3)^(2/3)*AppellF1[4/3, -2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])/(4*a*d*(1 + (b*x^3)/a)^(2/3))

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 )^{2/3}}{a d-b d x^3} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {x^3 \left (1+\frac {b x^3}{a}\right )^{2/3}}{a d-b d x^3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {x^4 \left (a+b x^3\right )^{2/3} F_1\left (\frac {4}{3};-\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{4 a d \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 216, normalized size = 0.94 \begin {gather*} \frac {\frac {15 x^4 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}+\frac {2^{2/3} a \left (\log \left (\frac {2^{2/3} b^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1\right )-2 \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )\right )}{b^{4/3}}-\frac {12 x \left (a+b x^3\right )^{2/3}}{b}}{36 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-12*x*(a + b*x^3)^(2/3))/b + (15*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/
a])/(a + b*x^3)^(1/3) + (2^(2/3)*a*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] -
2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*
b^(1/3)*x)/(b + a*x^3)^(1/3)]))/b^(4/3))/(36*d)

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IntegrateAlgebraic [A]  time = 0.84, size = 339, normalized size = 1.48 \begin {gather*} \frac {5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{9 b^{4/3} d}-\frac {2^{2/3} a \log \left (2^{2/3} \sqrt [3]{a+b x^3}-2 \sqrt [3]{b} x\right )}{3 b^{4/3} d}-\frac {5 a \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2 \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{3 \sqrt {3} b^{4/3} d}+\frac {2^{2/3} a \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{\sqrt {3} b^{4/3} d}-\frac {5 a \log \left (\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}+b^{2/3} x^2\right )}{18 b^{4/3} d}+\frac {a \log \left (2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}+2 b^{2/3} x^2\right )}{3 \sqrt [3]{2} b^{4/3} d}-\frac {x \left (a+b x^3\right )^{2/3}}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]

[Out]

-1/3*(x*(a + b*x^3)^(2/3))/(b*d) - (5*a*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))])/(3*Sqrt
[3]*b^(4/3)*d) + (2^(2/3)*a*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2^(2/3)*(a + b*x^3)^(1/3))])/(Sqrt[3]*b^(4
/3)*d) + (5*a*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(9*b^(4/3)*d) - (2^(2/3)*a*Log[-2*b^(1/3)*x + 2^(2/3)*(a
+ b*x^3)^(1/3)])/(3*b^(4/3)*d) - (5*a*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(18*
b^(4/3)*d) + (a*Log[2*b^(2/3)*x^2 + 2^(2/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(3*2^(1/
3)*b^(4/3)*d)

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fricas [A]  time = 0.51, size = 653, normalized size = 2.85 \begin {gather*} \left [-\frac {6 \cdot 4^{\frac {1}{3}} \sqrt {3} a b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{b}\right )^{\frac {1}{3}}}{3 \, x}\right ) - 15 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) - 6 \cdot 4^{\frac {1}{3}} a b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 3 \cdot 4^{\frac {1}{3}} a b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 4^{\frac {1}{3}} b x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x - 10 \, a b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 5 \, a b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{18 \, b^{2} d}, -\frac {6 \cdot 4^{\frac {1}{3}} \sqrt {3} a b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{b}\right )^{\frac {1}{3}}}{3 \, x}\right ) - 6 \cdot 4^{\frac {1}{3}} a b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 3 \cdot 4^{\frac {1}{3}} a b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 4^{\frac {1}{3}} b x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 30 \, \sqrt {\frac {1}{3}} a b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x - 10 \, a b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 5 \, a b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{18 \, b^{2} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/18*(6*4^(1/3)*sqrt(3)*a*b*(-1/b)^(1/3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/b)^(
1/3))/x) - 15*sqrt(1/3)*a*b*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt(1/3)*(b^(4
/3)*x^3 + (b*x^3 + a)^(1/3)*b*x^2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) - 6*4^(1/3)*a*b*(-1
/b)^(1/3)*log(-(4^(2/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 3*4^(1/3)*a*b*(-1/b)^(1/3)*log(-(2*4^(1/3
)*b*x^2*(-1/b)^(1/3) - 4^(2/3)*(b*x^3 + a)^(1/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(2/3))/x^2) + 6*(b*x^3 + a)^
(2/3)*b*x - 10*a*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) + 5*a*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^
(1/3)*b^(1/3)*x + (b*x^3 + a)^(2/3))/x^2))/(b^2*d), -1/18*(6*4^(1/3)*sqrt(3)*a*b*(-1/b)^(1/3)*arctan(-1/3*(sqr
t(3)*x - 4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/b)^(1/3))/x) - 6*4^(1/3)*a*b*(-1/b)^(1/3)*log(-(4^(2/3)*b*x*(-1
/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 3*4^(1/3)*a*b*(-1/b)^(1/3)*log(-(2*4^(1/3)*b*x^2*(-1/b)^(1/3) - 4^(2/3)*
(b*x^3 + a)^(1/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(2/3))/x^2) - 30*sqrt(1/3)*a*b^(2/3)*arctan(sqrt(1/3)*(b^(1
/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x)) + 6*(b*x^3 + a)^(2/3)*b*x - 10*a*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 +
a)^(1/3))/x) + 5*a*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^(2/3))/x^2))/(b^2*d)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{3}}{b d x^{3} - a d}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{3}}{-b d \,x^{3}+a d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{3}}{b d x^{3} - a d}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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