3.1776 \(\int \frac{\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx\)

Optimal. Leaf size=66 \[ \frac{36 b (a+b x)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac{6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(7/6))/(13*(b*c - a*d)*(c + d*x)^(13/6)) + (36*b*(a + b*x)^(7/6))/(91*(b*c - a*d)^2*(c + d*x)^(7/
6))

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Rubi [A]  time = 0.0090521, antiderivative size = 66, 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 = {45, 37} \[ \frac{36 b (a+b x)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac{6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(7/6))/(13*(b*c - a*d)*(c + d*x)^(13/6)) + (36*b*(a + b*x)^(7/6))/(91*(b*c - a*d)^2*(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)^{19/6}} \, dx &=\frac{6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac{(6 b) \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx}{13 (b c-a d)}\\ &=\frac{6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac{36 b (a+b x)^{7/6}}{91 (b c-a d)^2 (c+d x)^{7/6}}\\ \end{align*}

Mathematica [A]  time = 0.0220619, size = 46, normalized size = 0.7 \[ \frac{6 (a+b x)^{7/6} (-7 a d+13 b c+6 b d x)}{91 (c+d x)^{13/6} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(7/6)*(13*b*c - 7*a*d + 6*b*d*x))/(91*(b*c - a*d)^2*(c + d*x)^(13/6))

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Maple [A]  time = 0.005, size = 54, normalized size = 0.8 \begin{align*} -{\frac{-36\,bdx+42\,ad-78\,bc}{91\,{a}^{2}{d}^{2}-182\,abcd+91\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{13}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/6)/(d*x+c)^(19/6),x)

[Out]

-6/91*(b*x+a)^(7/6)*(-6*b*d*x+7*a*d-13*b*c)/(d*x+c)^(13/6)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

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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{19}{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)^(19/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(19/6), x)

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Fricas [B]  time = 1.55797, size = 373, normalized size = 5.65 \begin{align*} \frac{6 \,{\left (6 \, b^{2} d x^{2} + 13 \, a b c - 7 \, a^{2} d +{\left (13 \, b^{2} c - a b d\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{91 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{3} + 3 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2} + 3 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\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)^(19/6),x, algorithm="fricas")

[Out]

6/91*(6*b^2*d*x^2 + 13*a*b*c - 7*a^2*d + (13*b^2*c - a*b*d)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^2*c^5 - 2*a*
b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^3 + 3*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4
)*x^2 + 3*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*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((b*x+a)**(1/6)/(d*x+c)**(19/6),x)

[Out]

Timed out

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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)^(19/6),x, algorithm="giac")

[Out]

Timed out