3.1810 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac{432 b^2 (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^3}+\frac{72 b (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (72*b*(a + b*x)^(5/6))/(187*(b*c - a*d)^2*(c + d*x)^(1
1/6)) + (432*b^2*(a + b*x)^(5/6))/(935*(b*c - a*d)^3*(c + d*x)^(5/6))

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Rubi [A]  time = 0.0201421, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{432 b^2 (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^3}+\frac{72 b (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (72*b*(a + b*x)^(5/6))/(187*(b*c - a*d)^2*(c + d*x)^(1
1/6)) + (432*b^2*(a + b*x)^(5/6))/(935*(b*c - a*d)^3*(c + d*x)^(5/6))

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx &=\frac{6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac{(12 b) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{17 (b c-a d)}\\ &=\frac{6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac{72 b (a+b x)^{5/6}}{187 (b c-a d)^2 (c+d x)^{11/6}}+\frac{\left (72 b^2\right ) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{187 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac{72 b (a+b x)^{5/6}}{187 (b c-a d)^2 (c+d x)^{11/6}}+\frac{432 b^2 (a+b x)^{5/6}}{935 (b c-a d)^3 (c+d x)^{5/6}}\\ \end{align*}

Mathematica [A]  time = 0.033556, size = 77, normalized size = 0.76 \[ \frac{6 (a+b x)^{5/6} \left (55 a^2 d^2-10 a b d (17 c+6 d x)+b^2 \left (187 c^2+204 c d x+72 d^2 x^2\right )\right )}{935 (c+d x)^{17/6} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6)*(55*a^2*d^2 - 10*a*b*d*(17*c + 6*d*x) + b^2*(187*c^2 + 204*c*d*x + 72*d^2*x^2)))/(935*(b*c
- a*d)^3*(c + d*x)^(17/6))

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Maple [A]  time = 0.006, size = 105, normalized size = 1. \begin{align*} -{\frac{432\,{b}^{2}{d}^{2}{x}^{2}-360\,ab{d}^{2}x+1224\,{b}^{2}cdx+330\,{a}^{2}{d}^{2}-1020\,abcd+1122\,{b}^{2}{c}^{2}}{935\,{a}^{3}{d}^{3}-2805\,{a}^{2}cb{d}^{2}+2805\,a{b}^{2}{c}^{2}d-935\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{17}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x)

[Out]

-6/935*(b*x+a)^(5/6)*(72*b^2*d^2*x^2-60*a*b*d^2*x+204*b^2*c*d*x+55*a^2*d^2-170*a*b*c*d+187*b^2*c^2)/(d*x+c)^(1
7/6)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)

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Fricas [B]  time = 1.84383, size = 525, normalized size = 5.2 \begin{align*} \frac{6 \,{\left (72 \, b^{2} d^{2} x^{2} + 187 \, b^{2} c^{2} - 170 \, a b c d + 55 \, a^{2} d^{2} + 12 \,{\left (17 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{935 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3} +{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{3} + 3 \,{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} + 3 \,{\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x, algorithm="fricas")

[Out]

6/935*(72*b^2*d^2*x^2 + 187*b^2*c^2 - 170*a*b*c*d + 55*a^2*d^2 + 12*(17*b^2*c*d - 5*a*b*d^2)*x)*(b*x + a)^(5/6
)*(d*x + c)^(1/6)/(b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 +
3*a^2*b*c*d^5 - a^3*d^6)*x^3 + 3*(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^2 + 3*(b^3*c^
5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4)*x)

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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(1/(b*x+a)**(1/6)/(d*x+c)**(23/6),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)