3.1817 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx\)

Optimal. Leaf size=84 \[ \frac{6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{5}{6},\frac{19}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)^3} \]

[Out]

(6*b^2*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[5/6, 19/6, 11/6, -((d*(a + b*x))/(b
*c - a*d))])/(5*(b*c - a*d)^3*(c + d*x)^(1/6))

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Rubi [A]  time = 0.0204141, antiderivative size = 84, 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 = {70, 69} \[ \frac{6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{5}{6},\frac{19}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^2*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[5/6, 19/6, 11/6, -((d*(a + b*x))/(b
*c - a*d))])/(5*(b*c - a*d)^3*(c + d*x)^(1/6))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx &=\frac{\left (b^3 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}\right ) \int \frac{1}{\sqrt [6]{a+b x} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{19/6}} \, dx}{(b c-a d)^3 \sqrt [6]{c+d x}}\\ &=\frac{6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{5}{6},\frac{19}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 (b c-a d)^3 \sqrt [6]{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0328864, size = 81, normalized size = 0.96 \[ \frac{6 b (a+b x)^{5/6} \left (\frac{b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac{5}{6},\frac{19}{6};\frac{11}{6};\frac{d (a+b x)}{a d-b c}\right )}{5 (c+d x)^{7/6} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[5/6, 19/6, 11/6, (d*(a + b*x))/(-(b*c
) + a*d)])/(5*(b*c - a*d)^2*(c + d*x)^(7/6))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{19}{6}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{19}{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)^(19/6),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{b d^{4} x^{5} + a c^{4} +{\left (4 \, b c d^{3} + a d^{4}\right )} x^{4} + 2 \,{\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{3} + 2 \,{\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{2} +{\left (b c^{4} + 4 \, a c^{3} d\right )} x}, 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)^(19/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(b*d^4*x^5 + a*c^4 + (4*b*c*d^3 + a*d^4)*x^4 + 2*(3*b*c^2*d^2 + 2*a*c
*d^3)*x^3 + 2*(2*b*c^3*d + 3*a*c^2*d^2)*x^2 + (b*c^4 + 4*a*c^3*d)*x), 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)**(19/6),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{19}{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)^(19/6),x, algorithm="giac")

[Out]

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