3.1777 \(\int \frac{\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac{432 b^2 (a+b x)^{7/6}}{1729 (c+d x)^{7/6} (b c-a d)^3}+\frac{72 b (a+b x)^{7/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac{6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(7/6))/(19*(b*c - a*d)*(c + d*x)^(19/6)) + (72*b*(a + b*x)^(7/6))/(247*(b*c - a*d)^2*(c + d*x)^(1
3/6)) + (432*b^2*(a + b*x)^(7/6))/(1729*(b*c - a*d)^3*(c + d*x)^(7/6))

________________________________________________________________________________________

Rubi [A]  time = 0.0209129, 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} \[ \frac{432 b^2 (a+b x)^{7/6}}{1729 (c+d x)^{7/6} (b c-a d)^3}+\frac{72 b (a+b x)^{7/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac{6 (a+b x)^{7/6}}{19 (c+d x)^{19/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1/6)/(c + d*x)^(25/6),x]

[Out]

(6*(a + b*x)^(7/6))/(19*(b*c - a*d)*(c + d*x)^(19/6)) + (72*b*(a + b*x)^(7/6))/(247*(b*c - a*d)^2*(c + d*x)^(1
3/6)) + (432*b^2*(a + b*x)^(7/6))/(1729*(b*c - a*d)^3*(c + d*x)^(7/6))

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{\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx &=\frac{6 (a+b x)^{7/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac{(12 b) \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx}{19 (b c-a d)}\\ &=\frac{6 (a+b x)^{7/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac{72 b (a+b x)^{7/6}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac{\left (72 b^2\right ) \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx}{247 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{7/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac{72 b (a+b x)^{7/6}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac{432 b^2 (a+b x)^{7/6}}{1729 (b c-a d)^3 (c+d x)^{7/6}}\\ \end{align*}

Mathematica [A]  time = 0.0416167, size = 77, normalized size = 0.76 \[ \frac{6 (a+b x)^{7/6} \left (91 a^2 d^2-14 a b d (19 c+6 d x)+b^2 \left (247 c^2+228 c d x+72 d^2 x^2\right )\right )}{1729 (c+d x)^{19/6} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1/6)/(c + d*x)^(25/6),x]

[Out]

(6*(a + b*x)^(7/6)*(91*a^2*d^2 - 14*a*b*d*(19*c + 6*d*x) + b^2*(247*c^2 + 228*c*d*x + 72*d^2*x^2)))/(1729*(b*c
- a*d)^3*(c + d*x)^(19/6))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 105, normalized size = 1. \begin{align*} -{\frac{432\,{b}^{2}{d}^{2}{x}^{2}-504\,ab{d}^{2}x+1368\,{b}^{2}cdx+546\,{a}^{2}{d}^{2}-1596\,abcd+1482\,{b}^{2}{c}^{2}}{1729\,{a}^{3}{d}^{3}-5187\,{a}^{2}cb{d}^{2}+5187\,a{b}^{2}{c}^{2}d-1729\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{19}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/6)/(d*x+c)^(25/6),x)

[Out]

-6/1729*(b*x+a)^(7/6)*(72*b^2*d^2*x^2-84*a*b*d^2*x+228*b^2*c*d*x+91*a^2*d^2-266*a*b*c*d+247*b^2*c^2)/(d*x+c)^(
19/6)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{6}}}{{\left (d x + c\right )}^{\frac{25}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(25/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(25/6), x)

________________________________________________________________________________________

Fricas [B]  time = 1.50056, size = 695, normalized size = 6.88 \begin{align*} \frac{6 \,{\left (72 \, b^{3} d^{2} x^{3} + 247 \, a b^{2} c^{2} - 266 \, a^{2} b c d + 91 \, a^{3} d^{2} + 12 \,{\left (19 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} +{\left (247 \, b^{3} c^{2} - 38 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{1729 \,{\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3} +{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} x^{4} + 4 \,{\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} x^{3} + 6 \,{\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} x^{2} + 4 \,{\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(25/6),x, algorithm="fricas")

[Out]

6/1729*(72*b^3*d^2*x^3 + 247*a*b^2*c^2 - 266*a^2*b*c*d + 91*a^3*d^2 + 12*(19*b^3*c*d - a*b^2*d^2)*x^2 + (247*b
^3*c^2 - 38*a*b^2*c*d + 7*a^2*b*d^2)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5
*d^2 - a^3*c^4*d^3 + (b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*x^4 + 4*(b^3*c^4*d^3 - 3*a*b^2*
c^3*d^4 + 3*a^2*b*c^2*d^5 - a^3*c*d^6)*x^3 + 6*(b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)
*x^2 + 4*(b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4)*x)

________________________________________________________________________________________

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((b*x+a)**(1/6)/(d*x+c)**(25/6),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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((b*x+a)^(1/6)/(d*x+c)^(25/6),x, algorithm="giac")

[Out]

Timed out