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

Optimal. Leaf size=74 \[ \frac{2 (a+b x)^{3/2} \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{5},\frac{3}{2};\frac{5}{2};-\frac{d (a+b x)}{b c-a d}\right )}{3 b \sqrt [5]{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.0840637, antiderivative size = 74, 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 \[ \frac{2 (a+b x)^{3/2} \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{5},\frac{3}{2};\frac{5}{2};-\frac{d (a+b x)}{b c-a d}\right )}{3 b \sqrt [5]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7144, size = 61, normalized size = 0.82 \[ \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{4}{5}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{5} \\ \frac{9}{5} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{4 d \sqrt{\frac{d \left (a + b x\right )}{a d - b c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/(d*x+c)**(1/5),x)

[Out]

5*sqrt(a + b*x)*(c + d*x)**(4/5)*hyper((-1/2, 4/5), (9/5,), b*(-c - d*x)/(a*d -
b*c))/(4*d*sqrt(d*(a + b*x)/(a*d - b*c)))

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Mathematica [A]  time = 0.214534, size = 77, normalized size = 1.04 \[ \frac{5 \sqrt{a+b x} (c+d x)^{4/5} \left (\frac{5 \, _2F_1\left (\frac{1}{2},\frac{4}{5};\frac{9}{5};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+8\right )}{52 d} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[a + b*x]*(c + d*x)^(4/5)*(8 + (5*Hypergeometric2F1[1/2, 4/5, 9/5, (b*(c
+ d*x))/(b*c - a*d)])/Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]))/(52*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(1/5),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{1}{5}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(1/5),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(1/5),x, algorithm="giac")

[Out]

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