3.75 \(\int \frac{(c+d x^3)^2}{(a+b x^3)^{10/3}} \, dx\)

Optimal. Leaf size=78 \[ \frac{9 c^2 x}{14 a^3 \sqrt [3]{a+b x^3}}+\frac{3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac{x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}} \]

[Out]

(9*c^2*x)/(14*a^3*(a + b*x^3)^(1/3)) + (3*c*x*(c + d*x^3))/(14*a^2*(a + b*x^3)^(4/3)) + (x*(c + d*x^3)^2)/(7*a
*(a + b*x^3)^(7/3))

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Rubi [A]  time = 0.0211033, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {378, 191} \[ \frac{9 c^2 x}{14 a^3 \sqrt [3]{a+b x^3}}+\frac{3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac{x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(10/3),x]

[Out]

(9*c^2*x)/(14*a^3*(a + b*x^3)^(1/3)) + (3*c*x*(c + d*x^3))/(14*a^2*(a + b*x^3)^(4/3)) + (x*(c + d*x^3)^2)/(7*a
*(a + b*x^3)^(7/3))

Rule 378

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

Rule 191

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

Rubi steps

\begin{align*} \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{10/3}} \, dx &=\frac{x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{(6 c) \int \frac{c+d x^3}{\left (a+b x^3\right )^{7/3}} \, dx}{7 a}\\ &=\frac{3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac{x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{\left (9 c^2\right ) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{14 a^2}\\ &=\frac{9 c^2 x}{14 a^3 \sqrt [3]{a+b x^3}}+\frac{3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac{x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.080017, size = 126, normalized size = 1.62 \[ \frac{x \sqrt [3]{\frac{b x^3}{a}+1} \left (a^2 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+3 a b c x^3 \left (7 c+d x^3\right )+9 b^2 c^2 x^6\right )}{14 a^3 \left (a+b x^3\right )^{7/3} \sqrt [3]{\frac{d x^3}{c}+1} \sqrt [3]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(10/3),x]

[Out]

(x*(1 + (b*x^3)/a)^(1/3)*(9*b^2*c^2*x^6 + 3*a*b*c*x^3*(7*c + d*x^3) + a^2*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)))/(
14*a^3*(a + b*x^3)^(7/3)*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/3)*(1 + (d*x^3)/c)^(1/3))

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Maple [A]  time = 0.006, size = 76, normalized size = 1. \begin{align*}{\frac{x \left ( 2\,{a}^{2}{d}^{2}{x}^{6}+3\,abcd{x}^{6}+9\,{b}^{2}{c}^{2}{x}^{6}+7\,{a}^{2}cd{x}^{3}+21\,ab{c}^{2}{x}^{3}+14\,{a}^{2}{c}^{2} \right ) }{14\,{a}^{3}} \left ( b{x}^{3}+a \right ) ^{-{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^2/(b*x^3+a)^(10/3),x)

[Out]

1/14*x*(2*a^2*d^2*x^6+3*a*b*c*d*x^6+9*b^2*c^2*x^6+7*a^2*c*d*x^3+21*a*b*c^2*x^3+14*a^2*c^2)/(b*x^3+a)^(7/3)/a^3

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Maxima [A]  time = 0.959703, size = 147, normalized size = 1.88 \begin{align*} -\frac{{\left (4 \, b - \frac{7 \,{\left (b x^{3} + a\right )}}{x^{3}}\right )} c d x^{7}}{14 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2}} + \frac{d^{2} x^{7}}{7 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a} + \frac{{\left (2 \, b^{2} - \frac{7 \,{\left (b x^{3} + a\right )} b}{x^{3}} + \frac{14 \,{\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} c^{2} x^{7}}{14 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(10/3),x, algorithm="maxima")

[Out]

-1/14*(4*b - 7*(b*x^3 + a)/x^3)*c*d*x^7/((b*x^3 + a)^(7/3)*a^2) + 1/7*d^2*x^7/((b*x^3 + a)^(7/3)*a) + 1/14*(2*
b^2 - 7*(b*x^3 + a)*b/x^3 + 14*(b*x^3 + a)^2/x^6)*c^2*x^7/((b*x^3 + a)^(7/3)*a^3)

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Fricas [A]  time = 1.66892, size = 217, normalized size = 2.78 \begin{align*} \frac{{\left ({\left (9 \, b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{7} + 14 \, a^{2} c^{2} x + 7 \,{\left (3 \, a b c^{2} + a^{2} c d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{14 \,{\left (a^{3} b^{3} x^{9} + 3 \, a^{4} b^{2} x^{6} + 3 \, a^{5} b x^{3} + a^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(10/3),x, algorithm="fricas")

[Out]

1/14*((9*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*x^7 + 14*a^2*c^2*x + 7*(3*a*b*c^2 + a^2*c*d)*x^4)*(b*x^3 + a)^(2/3)/
(a^3*b^3*x^9 + 3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6)

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Sympy [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((d*x**3+c)**2/(b*x**3+a)**(10/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{10}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(10/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(10/3), x)