3.968 \(\int \frac{1}{(c x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{8 b^{5/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a+b x^2\right )^{3/4}}+\frac{4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}} \]

[Out]

(-2*(a + b*x^2)^(1/4))/(7*a*c*(c*x)^(7/2)) + (4*b*(a + b*x^2)^(1/4))/(7*a^2*c^3*
(c*x)^(3/2)) - (8*b^(5/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sq
rt[b]*x)/Sqrt[a]]/2, 2])/(7*a^(5/2)*c^6*(a + b*x^2)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.24402, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{8 b^{5/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a+b x^2\right )^{3/4}}+\frac{4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(9/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*(a + b*x^2)^(1/4))/(7*a*c*(c*x)^(7/2)) + (4*b*(a + b*x^2)^(1/4))/(7*a^2*c^3*
(c*x)^(3/2)) - (8*b^(5/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sq
rt[b]*x)/Sqrt[a]]/2, 2])/(7*a^(5/2)*c^6*(a + b*x^2)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.2175, size = 114, normalized size = 0.9 \[ - \frac{2 \sqrt [4]{a + b x^{2}}}{7 a c \left (c x\right )^{\frac{7}{2}}} + \frac{4 b \sqrt [4]{a + b x^{2}}}{7 a^{2} c^{3} \left (c x\right )^{\frac{3}{2}}} - \frac{8 b^{\frac{5}{2}} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{7 a^{\frac{5}{2}} c^{6} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(9/2)/(b*x**2+a)**(3/4),x)

[Out]

-2*(a + b*x**2)**(1/4)/(7*a*c*(c*x)**(7/2)) + 4*b*(a + b*x**2)**(1/4)/(7*a**2*c*
*3*(c*x)**(3/2)) - 8*b**(5/2)*(c*x)**(3/2)*(a/(b*x**2) + 1)**(3/4)*elliptic_f(at
an(sqrt(a)/(sqrt(b)*x))/2, 2)/(7*a**(5/2)*c**6*(a + b*x**2)**(3/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0818712, size = 92, normalized size = 0.73 \[ \frac{2 \sqrt{c x} \left (-a^2+4 b^2 x^4 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a b x^2+2 b^2 x^4\right )}{7 a^2 c^5 x^4 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(9/2)*(a + b*x^2)^(3/4)),x]

[Out]

(2*Sqrt[c*x]*(-a^2 + a*b*x^2 + 2*b^2*x^4 + 4*b^2*x^4*(1 + (b*x^2)/a)^(3/4)*Hyper
geometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(7*a^2*c^5*x^4*(a + b*x^2)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{9}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(9/2)/(b*x^2+a)^(3/4),x)

[Out]

int(1/(c*x)^(9/2)/(b*x^2+a)^(3/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(9/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(9/2)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(9/2)),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(3/4)*sqrt(c*x)*c^4*x^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(1/(c*x)**(9/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(9/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(9/2)), x)