3.991 \(\int \frac{(c x)^{13/2}}{\left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=155 \[ \frac{77 a^{5/2} c^6 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}}+\frac{77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac{11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}} \]

[Out]

(77*a^2*c^5*(c*x)^(3/2))/(60*b^3*(a + b*x^2)^(1/4)) - (11*a*c^3*(c*x)^(7/2))/(30
*b^2*(a + b*x^2)^(1/4)) + (c*(c*x)^(11/2))/(5*b*(a + b*x^2)^(1/4)) + (77*a^(5/2)
*c^6*(1 + a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2]
)/(20*b^(7/2)*(a + b*x^2)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.213203, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{77 a^{5/2} c^6 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}}+\frac{77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac{11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(13/2)/(a + b*x^2)^(5/4),x]

[Out]

(77*a^2*c^5*(c*x)^(3/2))/(60*b^3*(a + b*x^2)^(1/4)) - (11*a*c^3*(c*x)^(7/2))/(30
*b^2*(a + b*x^2)^(1/4)) + (c*(c*x)^(11/2))/(5*b*(a + b*x^2)^(1/4)) + (77*a^(5/2)
*c^6*(1 + a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2]
)/(20*b^(7/2)*(a + b*x^2)^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{77 a^{3} c^{6} \sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{40 b^{4} \sqrt [4]{a + b x^{2}}} + \frac{77 a^{2} c^{5} \left (c x\right )^{\frac{3}{2}}}{60 b^{3} \sqrt [4]{a + b x^{2}}} - \frac{11 a c^{3} \left (c x\right )^{\frac{7}{2}}}{30 b^{2} \sqrt [4]{a + b x^{2}}} + \frac{c \left (c x\right )^{\frac{11}{2}}}{5 b \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(13/2)/(b*x**2+a)**(5/4),x)

[Out]

77*a**3*c**6*sqrt(c*x)*(a/(b*x**2) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-5/4),
(x, 1/x))/(40*b**4*(a + b*x**2)**(1/4)) + 77*a**2*c**5*(c*x)**(3/2)/(60*b**3*(a
+ b*x**2)**(1/4)) - 11*a*c**3*(c*x)**(7/2)/(30*b**2*(a + b*x**2)**(1/4)) + c*(c*
x)**(11/2)/(5*b*(a + b*x**2)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.079182, size = 87, normalized size = 0.56 \[ \frac{c^5 (c x)^{3/2} \left (77 a^2 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-77 a^2-11 a b x^2+6 b^2 x^4\right )}{30 b^3 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(13/2)/(a + b*x^2)^(5/4),x]

[Out]

(c^5*(c*x)^(3/2)*(-77*a^2 - 11*a*b*x^2 + 6*b^2*x^4 + 77*a^2*(1 + (b*x^2)/a)^(1/4
)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(30*b^3*(a + b*x^2)^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{13}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(13/2)/(b*x^2+a)^(5/4),x)

[Out]

int((c*x)^(13/2)/(b*x^2+a)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{13}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(13/2)/(b*x^2 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((c*x)^(13/2)/(b*x^2 + a)^(5/4), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x} c^{6} x^{6}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(13/2)/(b*x^2 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(sqrt(c*x)*c^6*x^6/(b*x^2 + a)^(5/4), 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((c*x)**(13/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{13}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(13/2)/(b*x^2 + a)^(5/4),x, algorithm="giac")

[Out]

integrate((c*x)^(13/2)/(b*x^2 + a)^(5/4), x)