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

Optimal. Leaf size=344 \[ \frac{x \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right )}{15 d^2 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 b x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d} \]

[Out]

((8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*d^2*Sqrt[c + d*x^2
]) - (4*b*(b*c - 2*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (b*x*(a +
b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*d) - (Sqrt[c]*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*
d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(1
5*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(4*b
^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/S
qrt[c]], 1 - (b*c)/(a*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqr
t[c + d*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.728836, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{x \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right )}{15 d^2 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 b x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

((8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*d^2*Sqrt[c + d*x^2
]) - (4*b*(b*c - 2*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (b*x*(a +
b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*d) - (Sqrt[c]*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*
d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(1
5*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(4*b
^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/S
qrt[c]], 1 - (b*c)/(a*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqr
t[c + d*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 91.941, size = 328, normalized size = 0.95 \[ \frac{a^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (15 a^{2} d^{2} - 11 a b c d + 4 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 \sqrt{b} c d^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{\sqrt{a} \sqrt{b} \sqrt{c + d x^{2}} \left (23 a^{2} d^{2} - 23 a b c d + 8 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 d^{3} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{b x \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{5 d} + \frac{4 b x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{15 d^{2}} + \frac{b x \sqrt{c + d x^{2}} \left (23 a^{2} d^{2} - 23 a b c d + 8 b^{2} c^{2}\right )}{15 d^{3} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*sqrt(c + d*x**2)*(15*a**2*d**2 - 11*a*b*c*d + 4*b**2*c**2)*elliptic_f(a
tan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(15*sqrt(b)*c*d**2*sqrt(a*(c + d*x**2)/(
c*(a + b*x**2)))*sqrt(a + b*x**2)) - sqrt(a)*sqrt(b)*sqrt(c + d*x**2)*(23*a**2*d
**2 - 23*a*b*c*d + 8*b**2*c**2)*elliptic_e(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) +
1)/(15*d**3*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)) + b*x*(a +
b*x**2)**(3/2)*sqrt(c + d*x**2)/(5*d) + 4*b*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*
(2*a*d - b*c)/(15*d**2) + b*x*sqrt(c + d*x**2)*(23*a**2*d**2 - 23*a*b*c*d + 8*b*
*2*c**2)/(15*d**3*sqrt(a + b*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.845173, size = 260, normalized size = 0.76 \[ \frac{-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-34 a^2 b c d^2+27 a b^2 c^2 d-8 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+b d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (11 a d-4 b c+3 b d x^2\right )}{15 d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*b*c + 11*a*d + 3*b*d*x^2) - I*b*c*(
8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ell
ipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(-8*b^3*c^3 + 27*a*b^2*c^2*d - 3
4*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*
ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d
*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 615, normalized size = 1.8 \[{\frac{1}{15\,{d}^{3} \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) }\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{3}{d}^{3}+14\,\sqrt{-{\frac{b}{a}}}{x}^{5}a{b}^{2}{d}^{3}-\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{3}c{d}^{2}+11\,\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}b{d}^{3}+10\,\sqrt{-{\frac{b}{a}}}{x}^{3}a{b}^{2}c{d}^{2}-4\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{3}{c}^{2}d+15\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{3}{d}^{3}-34\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}bc{d}^{2}+27\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) a{b}^{2}{c}^{2}d-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{3}{c}^{3}+23\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}bc{d}^{2}-23\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) a{b}^{2}{c}^{2}d+8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{3}{c}^{3}+11\,\sqrt{-{\frac{b}{a}}}x{a}^{2}bc{d}^{2}-4\,\sqrt{-{\frac{b}{a}}}xa{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(3*(-b/a)^(1/2)*x^7*b^3*d^3+14*(-b/a)^(1/2)
*x^5*a*b^2*d^3-(-b/a)^(1/2)*x^5*b^3*c*d^2+11*(-b/a)^(1/2)*x^3*a^2*b*d^3+10*(-b/a
)^(1/2)*x^3*a*b^2*c*d^2-4*(-b/a)^(1/2)*x^3*b^3*c^2*d+15*((b*x^2+a)/a)^(1/2)*((d*
x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*d^3-34*((b*x^2+a)/
a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b*c*d
^2+27*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)
^(1/2))*a*b^2*c^2*d-8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)
^(1/2),(a*d/b/c)^(1/2))*b^3*c^3+23*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellip
ticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b*c*d^2-23*((b*x^2+a)/a)^(1/2)*((d*x^2+
c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^2*c^2*d+8*((b*x^2+a)/a
)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^3*c^3+11
*(-b/a)^(1/2)*x*a^2*b*c*d^2-4*(-b/a)^(1/2)*x*a*b^2*c^2*d)/d^3/(b*d*x^4+a*d*x^2+b
*c*x^2+a*c)/(-b/a)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**2)**(5/2)/sqrt(c + d*x**2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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