3.708 \(\int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+c x^2}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

[Out]

(-2*(Sqrt[-a] - Sqrt[c]*x)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/
((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^(1/4)*Hypergeometric2F1[-1/2
, 1/4, 1/2, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] -
Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*Sqrt[d + e*x]*(a + c*x^2)^(1/4))

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Rubi [A]  time = 0.234135, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a + c*x^2)^(1/4)),x]

[Out]

(-2*(Sqrt[-a] - Sqrt[c]*x)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/
((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^(1/4)*Hypergeometric2F1[-1/2
, 1/4, 1/2, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] -
Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*Sqrt[d + e*x]*(a + c*x^2)^(1/4))

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Rubi in Sympy [A]  time = 11.6014, size = 167, normalized size = 0.84 \[ - \frac{2 \sqrt [4]{\frac{\left (\sqrt{c} d + e \sqrt{- a}\right ) \left (\sqrt{c} x + \sqrt{- a}\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (\sqrt{c} x - \sqrt{- a}\right )}} \left (- \sqrt{c} x + \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{2 \sqrt{c} \sqrt{- a} \left (d + e x\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (- \sqrt{c} x + \sqrt{- a}\right )}} \right )}}{\sqrt [4]{a + c x^{2}} \sqrt{d + e x} \left (\sqrt{c} d + e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**(1/4),x)

[Out]

-2*((sqrt(c)*d + e*sqrt(-a))*(sqrt(c)*x + sqrt(-a))/((sqrt(c)*d - e*sqrt(-a))*(s
qrt(c)*x - sqrt(-a))))**(1/4)*(-sqrt(c)*x + sqrt(-a))*hyper((-1/2, 1/4), (1/2,),
2*sqrt(c)*sqrt(-a)*(d + e*x)/((sqrt(c)*d - e*sqrt(-a))*(-sqrt(c)*x + sqrt(-a)))
)/((a + c*x**2)**(1/4)*sqrt(d + e*x)*(sqrt(c)*d + e*sqrt(-a)))

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Mathematica [C]  time = 1.58723, size = 171, normalized size = 0.86 \[ \frac{2\ 2^{3/4} \left (\sqrt{c} x+i \sqrt{a}\right ) \sqrt [4]{\frac{\frac{i \sqrt{c} d x}{\sqrt{a}}-\frac{i \sqrt{a} e}{\sqrt{c}}+d+e x}{d+e x}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-\frac{i \sqrt{c} x d}{\sqrt{a}}+d+\frac{i \sqrt{a} e}{\sqrt{c}}+e x}{2 d+2 e x}\right )}{3 \sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{c} d-i \sqrt{a} e\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(1/4)),x]

[Out]

(2*2^(3/4)*(I*Sqrt[a] + Sqrt[c]*x)*((d - (I*Sqrt[a]*e)/Sqrt[c] + (I*Sqrt[c]*d*x)
/Sqrt[a] + e*x)/(d + e*x))^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (d + (I*Sqrt[a
]*e)/Sqrt[c] - (I*Sqrt[c]*d*x)/Sqrt[a] + e*x)/(2*d + 2*e*x)])/(3*(Sqrt[c]*d - I*
Sqrt[a]*e)*Sqrt[d + e*x]*(a + c*x^2)^(1/4))

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \[ \int{1 \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{c{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+a)^(1/4),x)

[Out]

int(1/(e*x+d)^(3/2)/(c*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(1/4)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(1/4)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(1/4)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((c*x^2 + a)^(1/4)*(e*x + d)^(3/2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{a + c x^{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**(1/4),x)

[Out]

Integral(1/((a + c*x**2)**(1/4)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(1/4)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + a)^(1/4)*(e*x + d)^(3/2)), x)