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

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 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]))

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])

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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 4.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {4}{5}}}{b^{2} d x^{3} + a^{2} c + {\left (b^{2} c + 2 \, a b d\right )} x^{2} + {\left (2 \, a b c + a^{2} d\right )} x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{5}}}\,{d x} \]

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)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {1}{5}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{5}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{1/5}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt [5]{c + d x}}\, dx \]

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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