3.16.90 \(\int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx\) [1590]

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

[Out]

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

________________________________________________________________________________________

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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} -\frac {3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x]

[Out]

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

Rule 37

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

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx &=-\frac {3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}}\\ \end {align*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.18, size = 27, normalized size = 0.84

method	result	size
gosper	\(\frac {3 \left (d x +c \right )^{\frac {2}{3}}}{2 \left (b x +a \right )^{\frac {2}{3}} \left (a d -b c \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.91, size = 42, normalized size = 1.31 \begin {gather*} -\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{3}} \sqrt [3]{c + d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(5/3)/(d*x+c)**(1/3),x)

[Out]

Integral(1/((a + b*x)**(5/3)*(c + d*x)**(1/3)), x)

________________________________________________________________________________________

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(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{5/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x)

[Out]

int(1/((a + b*x)^(5/3)*(c + d*x)^(1/3)), x)

________________________________________________________________________________________