3.1123 \(\int \frac{c+d x^2}{\sqrt{e x} \left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=116 \[ \frac{2 \sqrt{e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (a d+2 b c) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \sqrt{b} e^2 \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.275211, antiderivative size = 116, 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 \[ \frac{2 \sqrt{e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (a d+2 b c) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \sqrt{b} e^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(7/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.7218, size = 126, normalized size = 1.09 \[ - \frac{d \sqrt{e x}}{b e \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \sqrt{e x} \left (\frac{a d}{2} + b c\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{4 \left (e x\right )^{\frac{3}{2}} \left (\frac{a d}{2} + b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 a^{\frac{3}{2}} \sqrt{b} e^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

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)**(7/4),x)

[Out]

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

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Mathematica [C]  time = 0.0991147, size = 79, normalized size = 0.68 \[ \frac{2 x \left (\left (\frac{b x^2}{a}+1\right )^{3/4} (a d+2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-a d+b c\right )}{3 a b \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(7/4)),x]

[Out]

(2*x*(b*c - a*d + (2*b*c + a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4
, 5/4, -((b*x^2)/a)]))/(3*a*b*Sqrt[e*x]*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c){\frac{1}{\sqrt{ex}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(7/4),x)

[Out]

int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*sqrt(e*x)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*sqrt(e*x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*sqrt(e*x)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b*x^2 + a)^(7/4)*sqrt(e*x)), x)

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Sympy [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)/(e*x)**(1/2)/(b*x**2+a)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*sqrt(e*x)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*sqrt(e*x)), x)