3.60 \(\int \frac{c+d x^3}{(a+b x^3)^{7/3}} \, dx\)

Optimal. Leaf size=47 \[ \frac{3 c x}{4 a^2 \sqrt [3]{a+b x^3}}+\frac{x \left (c+d x^3\right )}{4 a \left (a+b x^3\right )^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.0097775, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {378, 191} \[ \frac{3 c x}{4 a^2 \sqrt [3]{a+b x^3}}+\frac{x \left (c+d x^3\right )}{4 a \left (a+b x^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)/(a + b*x^3)^(7/3),x]

[Out]

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

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

Mathematica [A]  time = 0.0202497, size = 37, normalized size = 0.79 \[ \frac{x \left (4 a c+a d x^3+3 b c x^3\right )}{4 a^2 \left (a+b x^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)/(a + b*x^3)^(7/3),x]

[Out]

(x*(4*a*c + 3*b*c*x^3 + a*d*x^3))/(4*a^2*(a + b*x^3)^(4/3))

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Maple [A]  time = 0.005, size = 34, normalized size = 0.7 \begin{align*}{\frac{x \left ( ad{x}^{3}+3\,bc{x}^{3}+4\,ac \right ) }{4\,{a}^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)/(b*x^3+a)^(7/3),x)

[Out]

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

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Maxima [A]  time = 0.957957, size = 69, normalized size = 1.47 \begin{align*} -\frac{{\left (b - \frac{4 \,{\left (b x^{3} + a\right )}}{x^{3}}\right )} c x^{4}}{4 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}} + \frac{d x^{4}}{4 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(7/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.66131, size = 117, normalized size = 2.49 \begin{align*} \frac{{\left ({\left (3 \, b c + a d\right )} x^{4} + 4 \, a c x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{4 \,{\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(7/3),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 96.6526, size = 190, normalized size = 4.04 \begin{align*} c \left (\frac{4 a x \Gamma \left (\frac{1}{3}\right )}{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{\frac{7}{3}} b x^{3} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right )} + \frac{3 b x^{4} \Gamma \left (\frac{1}{3}\right )}{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{\frac{7}{3}} b x^{3} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right )}\right ) + \frac{d x^{4} \Gamma \left (\frac{4}{3}\right )}{3 a^{\frac{7}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right ) + 3 a^{\frac{4}{3}} b x^{3} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)/(b*x**3+a)**(7/3),x)

[Out]

c*(4*a*x*gamma(1/3)/(9*a**(10/3)*(1 + b*x**3/a)**(1/3)*gamma(7/3) + 9*a**(7/3)*b*x**3*(1 + b*x**3/a)**(1/3)*ga
mma(7/3)) + 3*b*x**4*gamma(1/3)/(9*a**(10/3)*(1 + b*x**3/a)**(1/3)*gamma(7/3) + 9*a**(7/3)*b*x**3*(1 + b*x**3/
a)**(1/3)*gamma(7/3))) + d*x**4*gamma(4/3)/(3*a**(7/3)*(1 + b*x**3/a)**(1/3)*gamma(7/3) + 3*a**(4/3)*b*x**3*(1
+ b*x**3/a)**(1/3)*gamma(7/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(7/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)/(b*x^3 + a)^(7/3), x)