3.1734 \(\int \frac{1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0198808, 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} \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((b*(c + d*x))/(b*c - a*d))^(1/5)*Hypergeometric2F1[-1/2, 1/5, 1/2, -((d*(a + b*x))/(b*c - a*d))])/(b*Sqrt
[a + b*x]*(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}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx &=\frac{\sqrt [5]{\frac{b (c+d x)}{b c-a d}} \int \frac{1}{(a+b x)^{3/2} \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 [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.02144, size = 71, normalized size = 0.99 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\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)^(3/2)/(d*x+c)^(1/5),x)

[Out]

int(1/(b*x+a)^(3/2)/(d*x+c)^(1/5),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\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)^(3/2)/(d*x+c)^(1/5),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(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^{2} d x^{3} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} x^{2} +{\left (2 \, a b c + a^{2} d\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(1/5),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*(d*x + c)^(4/5)/(b^2*d*x^3 + a^2*c + (b^2*c + 2*a*b*d)*x^2 + (2*a*b*c + a^2*d)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt [5]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/5),x)

[Out]

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(1/5)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\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)^(3/2)/(d*x+c)^(1/5),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)), x)