3.1106 \(\int \frac{c+d x^2}{\sqrt{e x} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=122 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt{e}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt{e}}+\frac{2 \sqrt{e x} (b c-a d)}{a b e \sqrt [4]{a+b x^2}} \]

[Out]

(2*(b*c - a*d)*Sqrt[e*x])/(a*b*e*(a + b*x^2)^(1/4)) + (d*ArcTan[(b^(1/4)*Sqrt[e*
x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(b^(5/4)*Sqrt[e]) + (d*ArcTanh[(b^(1/4)*Sqrt[e
*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(b^(5/4)*Sqrt[e])

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Rubi [A]  time = 0.20268, antiderivative size = 122, 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 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt{e}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt{e}}+\frac{2 \sqrt{e x} (b c-a d)}{a b e \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(5/4)),x]

[Out]

(2*(b*c - a*d)*Sqrt[e*x])/(a*b*e*(a + b*x^2)^(1/4)) + (d*ArcTan[(b^(1/4)*Sqrt[e*
x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(b^(5/4)*Sqrt[e]) + (d*ArcTanh[(b^(1/4)*Sqrt[e
*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(b^(5/4)*Sqrt[e])

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Rubi in Sympy [A]  time = 26.3039, size = 110, normalized size = 0.9 \[ \frac{d \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{5}{4}} \sqrt{e}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{5}{4}} \sqrt{e}} - \frac{2 \sqrt{e x} \left (a d - b c\right )}{a b e \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(1/2)/(b*x**2+a)**(5/4),x)

[Out]

d*atan(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(b**(5/4)*sqrt(e)) + d*
atanh(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(b**(5/4)*sqrt(e)) - 2*s
qrt(e*x)*(a*d - b*c)/(a*b*e*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.0665856, size = 71, normalized size = 0.58 \[ \frac{2 x \left (a d \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-a d+b c\right )}{a b \sqrt{e x} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(5/4)),x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c){\frac{1}{\sqrt{ex}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(5/4),x)

[Out]

int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(5/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*sqrt(e*x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.250231, size = 482, normalized size = 3.95 \[ \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}}{\left (b c - a d\right )} \sqrt{e x} - 4 \,{\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac{d^{4}}{b^{5} e^{2}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (b^{2} e x^{2} + a b e\right )} \left (\frac{d^{4}}{b^{5} e^{2}}\right )^{\frac{1}{4}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d +{\left (b x^{2} + a\right )} \sqrt{\frac{\sqrt{b x^{2} + a} d^{2} e x +{\left (b^{3} e^{2} x^{2} + a b^{2} e^{2}\right )} \sqrt{\frac{d^{4}}{b^{5} e^{2}}}}{b x^{2} + a}}}\right ) +{\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac{d^{4}}{b^{5} e^{2}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d +{\left (b^{2} e x^{2} + a b e\right )} \left (\frac{d^{4}}{b^{5} e^{2}}\right )^{\frac{1}{4}}}{b x^{2} + a}\right ) -{\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac{d^{4}}{b^{5} e^{2}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d -{\left (b^{2} e x^{2} + a b e\right )} \left (\frac{d^{4}}{b^{5} e^{2}}\right )^{\frac{1}{4}}}{b x^{2} + a}\right )}{2 \,{\left (a b^{2} e x^{2} + a^{2} b e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*sqrt(e*x)),x, algorithm="fricas")

[Out]

1/2*(4*(b*x^2 + a)^(3/4)*(b*c - a*d)*sqrt(e*x) - 4*(a*b^2*e*x^2 + a^2*b*e)*(d^4/
(b^5*e^2))^(1/4)*arctan((b^2*e*x^2 + a*b*e)*(d^4/(b^5*e^2))^(1/4)/((b*x^2 + a)^(
3/4)*sqrt(e*x)*d + (b*x^2 + a)*sqrt((sqrt(b*x^2 + a)*d^2*e*x + (b^3*e^2*x^2 + a*
b^2*e^2)*sqrt(d^4/(b^5*e^2)))/(b*x^2 + a)))) + (a*b^2*e*x^2 + a^2*b*e)*(d^4/(b^5
*e^2))^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*d + (b^2*e*x^2 + a*b*e)*(d^4/(b^5*
e^2))^(1/4))/(b*x^2 + a)) - (a*b^2*e*x^2 + a^2*b*e)*(d^4/(b^5*e^2))^(1/4)*log(((
b*x^2 + a)^(3/4)*sqrt(e*x)*d - (b^2*e*x^2 + a*b*e)*(d^4/(b^5*e^2))^(1/4))/(b*x^2
+ a)))/(a*b^2*e*x^2 + a^2*b*e)

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Sympy [A]  time = 60.7568, size = 83, normalized size = 0.68 \[ \frac{c \Gamma \left (\frac{1}{4}\right )}{2 a \sqrt [4]{b} \sqrt{e} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{4}\right )} + \frac{d x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \sqrt{e} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)/(e*x)**(1/2)/(b*x**2+a)**(5/4),x)

[Out]

c*gamma(1/4)/(2*a*b**(1/4)*sqrt(e)*(a/(b*x**2) + 1)**(1/4)*gamma(5/4)) + d*x**(5
/2)*gamma(5/4)*hyper((5/4, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/4)*s
qrt(e)*gamma(9/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*sqrt(e*x)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*sqrt(e*x)), x)