3.1095 \(\int \frac{c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}} \]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(a*e*Sqrt[e*x]) - (d*ArcTan[(b^(1/4)*Sqrt[e*x])/(Sqrt[e
]*(a + b*x^2)^(1/4))])/(b^(3/4)*e^(3/2)) + (d*ArcTanh[(b^(1/4)*Sqrt[e*x])/(Sqrt[
e]*(a + b*x^2)^(1/4))])/(b^(3/4)*e^(3/2))

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Rubi [A]  time = 0.236559, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(a + b*x^2)^(1/4))/(a*e*Sqrt[e*x]) - (d*ArcTan[(b^(1/4)*Sqrt[e*x])/(Sqrt[e
]*(a + b*x^2)^(1/4))])/(b^(3/4)*e^(3/2)) + (d*ArcTanh[(b^(1/4)*Sqrt[e*x])/(Sqrt[
e]*(a + b*x^2)^(1/4))])/(b^(3/4)*e^(3/2))

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Rubi in Sympy [A]  time = 28.3293, size = 104, normalized size = 0.92 \[ - \frac{d \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{3}{4}} e^{\frac{3}{2}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{3}{4}} e^{\frac{3}{2}}} - \frac{2 c \sqrt [4]{a + b x^{2}}}{a e \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*atan(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(b**(3/4)*e**(3/2)) +
d*atanh(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(b**(3/4)*e**(3/2)) -
2*c*(a + b*x**2)**(1/4)/(a*e*sqrt(e*x))

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Mathematica [C]  time = 0.0712407, size = 77, normalized size = 0.68 \[ \frac{x \left (2 a d x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-6 c \left (a+b x^2\right )\right )}{3 a (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-6*c*(a + b*x^2) + 2*a*d*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[3/4, 3/
4, 7/4, -((b*x^2)/a)]))/(3*a*(e*x)^(3/2)*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 40.7813, size = 85, normalized size = 0.75 \[ \frac{\sqrt [4]{b} c \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{1}{4}\right )}{2 a e^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} + \frac{d x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(1/4)*c*(a/(b*x**2) + 1)**(1/4)*gamma(-1/4)/(2*a*e**(3/2)*gamma(3/4)) + d*x**
(3/2)*gamma(3/4)*hyper((3/4, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/4)
*e**(3/2)*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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