3.26.20 \(\int \frac {a+b x}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx\)

Optimal. Leaf size=32 \[ \frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {74} \begin {gather*} \frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} \frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b (c+2 d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 138.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(a + b*x)/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(4/3)), x]

________________________________________________________________________________________

fricas [A]  time = 1.51, size = 44, normalized size = 1.38 \begin {gather*} \frac {3 \, {\left (2 \, b d x + b c + a d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (2 \, b d^{3} x + b c d^{2} + a d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b x + a}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 27, normalized size = 0.84 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {2}{3}}}{2 \left (2 b d x +a d +b c \right )^{\frac {1}{3}} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.68, size = 26, normalized size = 0.81 \begin {gather*} \frac {3 \, {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (2 \, b d x + b c + a d\right )}^{\frac {1}{3}} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.05, size = 75, normalized size = 2.34 \begin {gather*} \frac {\left (\frac {3\,c}{4\,b\,d^3}+\frac {3\,x}{4\,b\,d^2}\right )\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{2/3}}{x\,{\left (c+d\,x\right )}^{1/3}+\frac {\left (2\,a\,d^3+2\,b\,c\,d^2\right )\,{\left (c+d\,x\right )}^{1/3}}{4\,b\,d^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x}{\sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac {4}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________