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

Optimal. Leaf size=410 \[ -\frac{2 x \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{35 b^2 d \sqrt{c+d x^2}}+\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{c^{3/2} \sqrt{a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{1}{35} x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (-\frac{2 a^2 d}{b}+9 a c+\frac{b c^2}{d}\right )+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{2 x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (4 b c-a d)}{35 b} \]

[Out]

(-2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(35*b^2*d*Sqr
t[c + d*x^2]) + ((9*a*c + (b*c^2)/d - (2*a^2*d)/b)*x*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])/35 + (2*(4*b*c - a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(35*b) + (d*x*(
a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*b) + (2*Sqrt[c]*(b*c + a*d)*(b^2*c^2 - 6*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^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
- (c^(3/2)*(b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(S
qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c +
d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.07085, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{2 x \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{35 b^2 d \sqrt{c+d x^2}}+\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{c^{3/2} \sqrt{a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{1}{35} x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (-\frac{2 a^2 d}{b}+9 a c+\frac{b c^2}{d}\right )+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{2 x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (4 b c-a d)}{35 b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(35*b^2*d*Sqr
t[c + d*x^2]) + ((9*a*c + (b*c^2)/d - (2*a^2*d)/b)*x*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])/35 + (2*(4*b*c - a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(35*b) + (d*x*(
a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*b) + (2*Sqrt[c]*(b*c + a*d)*(b^2*c^2 - 6*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^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
- (c^(3/2)*(b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(S
qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c +
d*x^2))]*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 141.37, size = 379, normalized size = 0.92 \[ - \frac{a^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a^{2} d^{2} - 18 a b c d + b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{35 b^{\frac{3}{2}} d \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{b x \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{\frac{5}{2}}}{7 d} + \frac{2 x \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d - b c\right )}{35 d} + \frac{x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a^{2} d^{2} + 9 a b c d - 2 b^{2} c^{2}\right )}{35 b d} + \frac{2 \sqrt{c} \sqrt{a + b x^{2}} \left (a d + b c\right ) \left (a^{2} d^{2} - 6 a b c d + b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{35 b^{2} d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{2 x \sqrt{a + b x^{2}} \left (a d + b c\right ) \left (a^{2} d^{2} - 6 a b c d + b^{2} c^{2}\right )}{35 b^{2} d \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)*sqrt(c + d*x**2)*(a**2*d**2 - 18*a*b*c*d + b**2*c**2)*elliptic_f(atan(
sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(35*b**(3/2)*d*sqrt(a*(c + d*x**2)/(c*(a + b
*x**2)))*sqrt(a + b*x**2)) + b*x*sqrt(a + b*x**2)*(c + d*x**2)**(5/2)/(7*d) + 2*
x*sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(4*a*d - b*c)/(35*d) + x*sqrt(a + b*x**2)
*sqrt(c + d*x**2)*(a**2*d**2 + 9*a*b*c*d - 2*b**2*c**2)/(35*b*d) + 2*sqrt(c)*sqr
t(a + b*x**2)*(a*d + b*c)*(a**2*d**2 - 6*a*b*c*d + b**2*c**2)*elliptic_e(atan(sq
rt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(35*b**2*d**(3/2)*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 + b*c)*(a**2*d**2 - 6*a*b
*c*d + b**2*c**2)/(35*b**2*d*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.06385, size = 302, normalized size = 0.74 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (17 c+8 d x^2\right )+b^2 \left (c^2+8 c d x^2+5 d^2 x^4\right )\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^3 d^3+8 a^2 b c d^2-11 a b^2 c^2 d+2 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^3 d^3-5 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{35 b d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a^2*d^2 + a*b*d*(17*c + 8*d*x^2) + b^2*(
c^2 + 8*c*d*x^2 + 5*d^2*x^4)) + (2*I)*c*(b^3*c^3 - 5*a*b^2*c^2*d - 5*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[Sqrt[b/a
]*x], (a*d)/(b*c)] - I*c*(2*b^3*c^3 - 11*a*b^2*c^2*d + 8*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^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.028, size = 780, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2),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+13*(-b/a)^(1/2)*x^7*b^3*c*d^3+9*(-b/a)^(1/2)*x^5*a^2*b*d^4+38*(-b
/a)^(1/2)*x^5*a*b^2*c*d^3+9*(-b/a)^(1/2)*x^5*b^3*c^2*d^2+(-b/a)^(1/2)*x^3*a^3*d^
4+26*(-b/a)^(1/2)*x^3*a^2*b*c*d^3+26*(-b/a)^(1/2)*x^3*a*b^2*c^2*d^2+(-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+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))*a^2*b*c^2*d^2-11*((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+2*((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+10*((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+10*((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-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))*b^3*c^4+(-b/a)^(1/2)*x*a^3*c*d^3
+17*(-b/a)^(1/2)*x*a^2*b*c^2*d^2+(-b/a)^(1/2)*x*a*b^2*c^3*d)/b/d^2/(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. \[ \int{\left (b x^{2} + a\right )}^{\frac{3}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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