3.1733 \(\int \frac{1}{\sqrt{a+b x} \sqrt [5]{c+d x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{2 \sqrt{a+b x} \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{5},\frac{1}{2};\frac{3}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [5]{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.0196033, antiderivative size = 72, 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} \[ \frac{2 \sqrt{a+b x} \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{5},\frac{1}{2};\frac{3}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [5]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0205918, size = 71, normalized size = 0.99 \[ \frac{2 \sqrt{a+b x} \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{5},\frac{1}{2};\frac{3}{2};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt [5]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt [5]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/2)/(d*x+c)^(1/5),x)

[Out]

int(1/(b*x+a)^(1/2)/(d*x+c)^(1/5),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(1/5),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{4}{5}}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(1/5),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*(d*x + c)^(4/5)/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x} \sqrt [5]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/5),x)

[Out]

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

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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(1/(b*x+a)^(1/2)/(d*x+c)^(1/5),x, algorithm="giac")

[Out]

Timed out