3.6.93 \(\int \frac {x^6 (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\) [593]

Optimal. Leaf size=264 \[ -\frac {4 a x \left (a+b x^3\right )^{2/3}}{9 b^2 d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}-\frac {14 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} d}+\frac {2^{2/3} a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3} d}+\frac {a^2 \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{7/3} d}-\frac {a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{7/3} d}+\frac {7 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{9 b^{7/3} d} \]

[Out]

-4/9*a*x*(b*x^3+a)^(2/3)/b^2/d-1/6*x^4*(b*x^3+a)^(2/3)/b/d+1/6*a^2*ln(-b*d*x^3+a*d)*2^(2/3)/b^(7/3)/d-1/2*a^2*
ln(2^(1/3)*b^(1/3)*x-(b*x^3+a)^(1/3))*2^(2/3)/b^(7/3)/d+7/9*a^2*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(7/3)/d-14/27
*a^2*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(7/3)/d*3^(1/2)+1/3*2^(2/3)*a^2*arctan(1/3*(1+2*2^(
1/3)*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(7/3)/d*3^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {489, 596, 544, 245, 384} \begin {gather*} -\frac {14 a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} d}+\frac {2^{2/3} a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3} d}+\frac {a^2 \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{7/3} d}-\frac {a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{7/3} d}+\frac {7 a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{9 b^{7/3} d}-\frac {4 a x \left (a+b x^3\right )^{2/3}}{9 b^2 d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*x*(a + b*x^3)^(2/3))/(9*b^2*d) - (x^4*(a + b*x^3)^(2/3))/(6*b*d) - (14*a^2*ArcTan[(1 + (2*b^(1/3)*x)/(a
+ b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(7/3)*d) + (2^(2/3)*a^2*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(a + b*x^3)^(
1/3))/Sqrt[3]])/(Sqrt[3]*b^(7/3)*d) + (a^2*Log[a*d - b*d*x^3])/(3*2^(1/3)*b^(7/3)*d) - (a^2*Log[2^(1/3)*b^(1/3
)*x - (a + b*x^3)^(1/3)])/(2^(1/3)*b^(7/3)*d) + (7*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(9*b^(7/3)*d)

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 489

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

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rubi steps

\begin {align*} \int \frac {x^6 \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^6 \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^7 \left (a+b x^3\right )^{2/3} F_1\left (\frac {7}{3};-\frac {2}{3},1;\frac {10}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{7 a d \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 325, normalized size = 1.23 \begin {gather*} -\frac {24 a \sqrt [3]{b} x \left (a+b x^3\right )^{2/3}+9 b^{4/3} x^4 \left (a+b x^3\right )^{2/3}+28 \sqrt {3} a^2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-18\ 2^{2/3} \sqrt {3} a^2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )-28 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+18\ 2^{2/3} a^2 \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )+14 a^2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-9\ 2^{2/3} a^2 \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{54 b^{7/3} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*(24*a*b^(1/3)*x*(a + b*x^3)^(2/3) + 9*b^(4/3)*x^4*(a + b*x^3)^(2/3) + 28*Sqrt[3]*a^2*ArcTan[(Sqrt[3]*b^(
1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 18*2^(2/3)*Sqrt[3]*a^2*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2^
(2/3)*(a + b*x^3)^(1/3))] - 28*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] + 18*2^(2/3)*a^2*Log[-2*b^(1/3)*x + 2
^(2/3)*(a + b*x^3)^(1/3)] + 14*a^2*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 9*2^(2
/3)*a^2*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)])/(b^(7/3)*d)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{6} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b d \,x^{3}+a d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^6*(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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 6.43, size = 701, normalized size = 2.66 \begin {gather*} \left [-\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a^{2} 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 ) - 42 \, \sqrt {\frac {1}{3}} a^{2} 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 ) - 18 \cdot 4^{\frac {1}{3}} a^{2} 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 ) + 9 \cdot 4^{\frac {1}{3}} a^{2} 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 ) - 28 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 14 \, a^{2} 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 ) + 3 \, {\left (3 \, b^{2} x^{4} + 8 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3} d}, -\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a^{2} 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 ) - 18 \cdot 4^{\frac {1}{3}} a^{2} 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 ) + 9 \cdot 4^{\frac {1}{3}} a^{2} 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 ) - 84 \, \sqrt {\frac {1}{3}} a^{2} 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 ) - 28 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 14 \, a^{2} 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 ) + 3 \, {\left (3 \, b^{2} x^{4} + 8 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/54*(18*4^(1/3)*sqrt(3)*a^2*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) - 42*sqrt(1/3)*a^2*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) - 18*4^(1/3)*a
^2*b*(-1/b)^(1/3)*log(-(4^(2/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 9*4^(1/3)*a^2*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) - 28*a
^2*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) + 14*a^2*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) + 3*(3*b^2*x^4 + 8*a*b*x)*(b*x^3 + a)^(2/3))/(b^3*d), -1/54*(18*4^(1/3)*sqrt(3)
*a^2*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) - 18*4^(1/3)*a
^2*b*(-1/b)^(1/3)*log(-(4^(2/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 9*4^(1/3)*a^2*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) - 84*s
qrt(1/3)*a^2*b^(2/3)*arctan(sqrt(1/3)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x)) - 28*a^2*b^(2/3)*log(-(b^
(1/3)*x - (b*x^3 + a)^(1/3))/x) + 14*a^2*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) + 3*(3*b^2*x^4 + 8*a*b*x)*(b*x^3 + a)^(2/3))/(b^3*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {x^{6} \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**6*(b*x**3+a)**(2/3)/(-b*d*x**3+a*d),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(b*x^3 + a)^(2/3)*x^6/(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^6\,{\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^6*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x)

[Out]

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

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