3.1094 \(\int \frac{\sqrt{e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{\sqrt{e} (4 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac{\sqrt{e} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac{d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e} \]

[Out]

(d*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(2*b*e) - ((4*b*c - 3*a*d)*Sqrt[e]*ArcTan[(b^(
1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(7/4)) + ((4*b*c - 3*a*d)*Sqr
t[e]*ArcTanh[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(7/4))

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Rubi [A]  time = 0.270842, antiderivative size = 136, 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 \[ -\frac{\sqrt{e} (4 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac{\sqrt{e} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac{d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e} \]

Antiderivative was successfully verified.

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

[Out]

(d*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(2*b*e) - ((4*b*c - 3*a*d)*Sqrt[e]*ArcTan[(b^(
1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(7/4)) + ((4*b*c - 3*a*d)*Sqr
t[e]*ArcTanh[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(7/4))

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Rubi in Sympy [A]  time = 29.1738, size = 124, normalized size = 0.91 \[ \frac{d \left (e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}}}{2 b e} + \frac{\sqrt{e} \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{7}{4}}} - \frac{\sqrt{e} \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(e*x)**(3/2)*(a + b*x**2)**(1/4)/(2*b*e) + sqrt(e)*(3*a*d - 4*b*c)*atan(b**(1/
4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(4*b**(7/4)) - sqrt(e)*(3*a*d - 4*b*
c)*atanh(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(4*b**(7/4))

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Mathematica [C]  time = 0.0974838, size = 80, normalized size = 0.59 \[ \frac{x \sqrt{e x} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} (4 b c-3 a d) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+3 d \left (a+b x^2\right )\right )}{6 b \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[e*x]*(3*d*(a + b*x^2) + (4*b*c - 3*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeome
tric2F1[3/4, 3/4, 7/4, -((b*x^2)/a)]))/(6*b*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c)\sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x)^(1/2)*(d*x^2+c)/(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)*sqrt(e*x)/(b*x^2 + a)^(3/4),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)*sqrt(e*x)/(b*x^2 + a)^(3/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 5.8518, size = 92, normalized size = 0.68 \[ \frac{c \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} e \Gamma \left (\frac{7}{4}\right )} + \frac{d \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} e^{3} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(e*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*gamma(7/4)) + d*(e*x)**(7/2)*gamma(7/4)*hyper((3/4, 7/4), (11/4,), b
*x**2*exp_polar(I*pi)/a)/(2*a**(3/4)*e**3*gamma(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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