3.57 \(\int (a+b x^3)^{2/3} (c+d x^3) \, dx\)

Optimal. Leaf size=141 \[ -\frac{a (6 b c-a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}+\frac{a (6 b c-a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{x \left (a+b x^3\right )^{2/3} (6 b c-a d)}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b} \]

[Out]

((6*b*c - a*d)*x*(a + b*x^3)^(2/3))/(18*b) + (d*x*(a + b*x^3)^(5/3))/(6*b) + (a*(6*b*c - a*d)*ArcTan[(1 + (2*b
^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(4/3)) - (a*(6*b*c - a*d)*Log[-(b^(1/3)*x) + (a + b*x^3)^(
1/3)])/(18*b^(4/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0450737, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {388, 195, 239} \[ -\frac{a (6 b c-a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}+\frac{a (6 b c-a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{x \left (a+b x^3\right )^{2/3} (6 b c-a d)}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3)*(c + d*x^3),x]

[Out]

((6*b*c - a*d)*x*(a + b*x^3)^(2/3))/(18*b) + (d*x*(a + b*x^3)^(5/3))/(6*b) + (a*(6*b*c - a*d)*ArcTan[(1 + (2*b
^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(4/3)) - (a*(6*b*c - a*d)*Log[-(b^(1/3)*x) + (a + b*x^3)^(
1/3)])/(18*b^(4/3))

Rule 388

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 239

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]

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right ) \, dx &=\frac{d x \left (a+b x^3\right )^{5/3}}{6 b}-\frac{(-6 b c+a d) \int \left (a+b x^3\right )^{2/3} \, dx}{6 b}\\ &=\frac{(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac{(a (6 b c-a d)) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx}{9 b}\\ &=\frac{(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac{a (6 b c-a d) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}-\frac{a (6 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.0694077, size = 72, normalized size = 0.51 \[ \frac{x \left (a+b x^3\right )^{2/3} \left (\frac{(6 b c-a d) \, _2F_1\left (-\frac{2}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (\frac{b x^3}{a}+1\right )^{2/3}}+d \left (a+b x^3\right )\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x^3)^(2/3)*(d*(a + b*x^3) + ((6*b*c - a*d)*Hypergeometric2F1[-2/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*
x^3)/a)^(2/3)))/(6*b)

________________________________________________________________________________________

Maple [F]  time = 0.222, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}} \left ( d{x}^{3}+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.02218, size = 1087, normalized size = 7.71 \begin{align*} \left [-\frac{3 \, \sqrt{\frac{1}{3}}{\left (6 \, a b^{2} c - a^{2} b d\right )} \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 ) + 2 \,{\left (6 \, a b c - a^{2} d\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (6 \, a b c - a^{2} d\right )} 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} d x^{4} + 2 \,{\left (3 \, b^{2} c + a b d\right )} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{54 \, b^{2}}, -\frac{2 \,{\left (6 \, a b c - a^{2} d\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (6 \, a b c - a^{2} d\right )} 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 ) + \frac{6 \, \sqrt{\frac{1}{3}}{\left (6 \, a b^{2} c - a^{2} b d\right )} \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 )}{b^{\frac{1}{3}}} - 3 \,{\left (3 \, b^{2} d x^{4} + 2 \,{\left (3 \, b^{2} c + a b d\right )} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{54 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/54*(3*sqrt(1/3)*(6*a*b^2*c - a^2*b*d)*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*s
qrt(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) + 2*(
6*a*b*c - a^2*d)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - (6*a*b*c - a^2*d)*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*d*x^4 + 2*(3*b^2*c + a*b*d)*x)*(b*x^3 + a)^
(2/3))/b^2, -1/54*(2*(6*a*b*c - a^2*d)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - (6*a*b*c - a^2*d)*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) + 6*sqrt(1/3)*(6*a*b^2*c - a^2*b
*d)*arctan(sqrt(1/3)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x))/b^(1/3) - 3*(3*b^2*d*x^4 + 2*(3*b^2*c + a*
b*d)*x)*(b*x^3 + a)^(2/3))/b^2]

________________________________________________________________________________________

Sympy [C]  time = 3.2304, size = 82, normalized size = 0.58 \begin{align*} \frac{a^{\frac{2}{3}} c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{a^{\frac{2}{3}} d x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(2/3)*c*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + a**(2/3)*d*x**4*
gamma(4/3)*hyper((-2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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