3.950 \(\int \frac{x^4 \left (a+b x^2\right )^{3/2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=429 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2} (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right )}{35 b d^3}-\frac{2 x \sqrt{a+b x^2} (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right )}{35 b^2 d^3 \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-4 a d)}{35 d^2}+\frac{b x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}}{7 d} \]

[Out]

(-2*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*x*Sqrt[a + b*x^2])/(35*b^2*d
^3*Sqrt[c + d*x^2]) + ((8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt
[c + d*x^2])/(35*b*d^3) - (2*(3*b*c - 4*a*d)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]
)/(35*d^2) + (b*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) + (2*Sqrt[c]*(2*b*c -
a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d
]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^2*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*
x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*Sqrt[a + b
*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b*d^(7/2)*Sqr
t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.40105, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2} (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right )}{35 b d^3}-\frac{2 x \sqrt{a+b x^2} (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right )}{35 b^2 d^3 \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-4 a d)}{35 d^2}+\frac{b x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}}{7 d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*x*Sqrt[a + b*x^2])/(35*b^2*d
^3*Sqrt[c + d*x^2]) + ((8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt
[c + d*x^2])/(35*b*d^3) - (2*(3*b*c - 4*a*d)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]
)/(35*d^2) + (b*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) + (2*Sqrt[c]*(2*b*c -
a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d
]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^2*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*
x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*Sqrt[a + b
*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b*d^(7/2)*Sqr
t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 162.498, size = 398, normalized size = 0.93 \[ \frac{b x^{5} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{7 d} + \frac{2 x^{3} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (4 a d - 3 b c\right )}{35 d^{2}} - \frac{c^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (a^{2} d^{2} - 11 a b c d + 8 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{35 b d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a^{2} d^{2} - 11 a b c d + 8 b^{2} c^{2}\right )}{35 b d^{3}} + \frac{2 \sqrt{c} \sqrt{a + b x^{2}} \left (a d - 2 b c\right ) \left (a^{2} d^{2} + 4 a b c d - 4 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{35 b^{2} d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{2 x \sqrt{a + b x^{2}} \left (a d - 2 b c\right ) \left (a^{2} d^{2} + 4 a b c d - 4 b^{2} c^{2}\right )}{35 b^{2} d^{3} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**5*sqrt(a + b*x**2)*sqrt(c + d*x**2)/(7*d) + 2*x**3*sqrt(a + b*x**2)*sqrt(c
+ d*x**2)*(4*a*d - 3*b*c)/(35*d**2) - c**(3/2)*sqrt(a + b*x**2)*(a**2*d**2 - 11*
a*b*c*d + 8*b**2*c**2)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(35*b*
d**(7/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) + x*sqrt(a + b*
x**2)*sqrt(c + d*x**2)*(a**2*d**2 - 11*a*b*c*d + 8*b**2*c**2)/(35*b*d**3) + 2*sq
rt(c)*sqrt(a + b*x**2)*(a*d - 2*b*c)*(a**2*d**2 + 4*a*b*c*d - 4*b**2*c**2)*ellip
tic_e(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(35*b**2*d**(7/2)*sqrt(c*(a + b*x*
*2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) - 2*x*sqrt(a + b*x**2)*(a*d - 2*b*c)*(a*
*2*d**2 + 4*a*b*c*d - 4*b**2*c**2)/(35*b**2*d**3*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.09909, size = 305, normalized size = 0.71 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (8 d x^2-11 c\right )+b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^3 d^3+15 a^2 b c d^2-32 a b^2 c^2 d+16 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^3 d^3+2 a^2 b c d^2-12 a b^2 c^2 d+8 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{35 b d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a^2*d^2 + a*b*d*(-11*c + 8*d*x^2) + b^2*
(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) + (2*I)*c*(8*b^3*c^3 - 12*a*b^2*c^2*d + 2*a^2*b
*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sq
rt[b/a]*x], (a*d)/(b*c)] - I*c*(16*b^3*c^3 - 32*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 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)])/(35*b*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.029, size = 782, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(5*(-b/a)^(1/2)*x^9*b^3*d^4+13*(-b/a)^(1/2)
*x^7*a*b^2*d^4-(-b/a)^(1/2)*x^7*b^3*c*d^3+9*(-b/a)^(1/2)*x^5*a^2*b*d^4-4*(-b/a)^
(1/2)*x^5*a*b^2*c*d^3+2*(-b/a)^(1/2)*x^5*b^3*c^2*d^2+(-b/a)^(1/2)*x^3*a^3*d^4-2*
(-b/a)^(1/2)*x^3*a^2*b*c*d^3-9*(-b/a)^(1/2)*x^3*a*b^2*c^2*d^2+8*(-b/a)^(1/2)*x^3
*b^3*c^3*d+((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*c*d^3+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^2*b*c^2*d^2-32*((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^3*d+16*((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^4-2*((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^3*
c*d^3-4*((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^2*b*c^2*d^2+24*((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^3*d-16*((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^4+(-b/a)^(1/2)*x*a^3*c*d^3-1
1*(-b/a)^(1/2)*x*a^2*b*c^2*d^2+8*(-b/a)^(1/2)*x*a*b^2*c^3*d)/b/d^4/(b*d*x^4+a*d*
x^2+b*c*x^2+a*c)/(-b/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)*x^4/sqrt(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*(a + b*x**2)**(3/2)/sqrt(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)*x^4/sqrt(d*x^2 + c), x)