3.1135 \(\int \frac{\sqrt{e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 (e x)^{3/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (3 a d+2 b c) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} b^{3/2} \sqrt [4]{a+b x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.185584, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 (e x)^{3/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (3 a d+2 b c) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} b^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d \left (e x\right )^{\frac{3}{2}}}{b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (\frac{3 a d}{2} + b c\right )}{5 a b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \sqrt{e x} \left (\frac{3 a d}{2} + b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{5 a b^{2} \sqrt [4]{a + b x^{2}}} - \frac{4 \sqrt{e x} \left (\frac{3 a d}{2} + b c\right )}{5 a b^{2} x \sqrt [4]{a + b x^{2}}} \]

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)**(9/4),x)

[Out]

-d*(e*x)**(3/2)/(b*e*(a + b*x**2)**(5/4)) + 2*(e*x)**(3/2)*(3*a*d/2 + b*c)/(5*a*
b*e*(a + b*x**2)**(5/4)) + 2*sqrt(e*x)*(3*a*d/2 + b*c)*(a/(b*x**2) + 1)**(1/4)*I
ntegral((a*x**2/b + 1)**(-1/4), (x, 1/x))/(5*a*b**2*(a + b*x**2)**(1/4)) - 4*sqr
t(e*x)*(3*a*d/2 + b*c)/(5*a*b**2*x*(a + b*x**2)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.157829, size = 111, normalized size = 0.97 \[ -\frac{2 \sqrt{e x} \left (2 x \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (3 a d+2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-3 x \left (2 a^2 d+3 a b \left (c+d x^2\right )+2 b^2 c x^2\right )\right )}{15 a^2 b \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(-2*Sqrt[e*x]*(-3*x*(2*a^2*d + 2*b^2*c*x^2 + 3*a*b*(c + d*x^2)) + 2*(2*b*c + 3*a
*d)*x*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^
2)/a)]))/(15*a^2*b*(a + b*x^2)^(5/4))

_______________________________________________________________________________________

Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c)\sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

[Out]

int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4), x)