3.210 \(\int \frac{1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=323 \[ \frac{\sqrt{d} \sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 b x (b c-3 a d)}{3 a^2 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}-\frac{b \sqrt{c} \sqrt{d} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d)} \]

[Out]

(b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]) + (2*b*(b*c - 3*a*d)*x
)/(3*a^2*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[d]*(2*b^2*c^2 -
7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1
- (b*c)/(a*d)])/(3*a^2*Sqrt[c]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2)
)]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*Sqrt[d]*(b*c - 9*a*d)*Sqrt[a + b*x^2]*EllipticF
[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*(b*c - a*d)^3*Sqrt[(c*(a
+ b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.674622, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\sqrt{d} \sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 b x (b c-3 a d)}{3 a^2 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}-\frac{b \sqrt{c} \sqrt{d} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]

[Out]

(b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]) + (2*b*(b*c - 3*a*d)*x
)/(3*a^2*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[d]*(2*b^2*c^2 -
7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1
- (b*c)/(a*d)])/(3*a^2*Sqrt[c]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2)
)]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*Sqrt[d]*(b*c - 9*a*d)*Sqrt[a + b*x^2]*EllipticF
[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*(b*c - a*d)^3*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 = 96.8982, size = 284, normalized size = 0.88 \[ \frac{d x}{c \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{b x \sqrt{c + d x^{2}} \left (3 a d + b c\right )}{3 a c \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{b \sqrt{c} \sqrt{d} \sqrt{a + b x^{2}} \left (9 a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 a^{2} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} + \frac{\sqrt{b} \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 7 a b c d - 2 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 a^{\frac{3}{2}} c \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)

[Out]

d*x/(c*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(a*d - b*c)) + b*x*sqrt(c + d*x**2)*
(3*a*d + b*c)/(3*a*c*(a + b*x**2)**(3/2)*(a*d - b*c)**2) - b*sqrt(c)*sqrt(d)*sqr
t(a + b*x**2)*(9*a*d - b*c)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(
3*a**2*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)*(a*d - b*c)**3) +
sqrt(b)*sqrt(c + d*x**2)*(3*a**2*d**2 + 7*a*b*c*d - 2*b**2*c**2)*elliptic_e(atan
(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(3*a**(3/2)*c*sqrt(a*(c + d*x**2)/(c*(a + b
*x**2)))*sqrt(a + b*x**2)*(a*d - b*c)**3)

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Mathematica [C]  time = 1.84406, size = 337, normalized size = 1.04 \[ \frac{2 i b c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+x \sqrt{\frac{b}{a}} \left (3 a^4 d^3+6 a^3 b d^3 x^2+a^2 b^2 d \left (8 c^2+8 c d x^2+3 d^2 x^4\right )+a b^3 c \left (-3 c^2+4 c d x^2+7 d^2 x^4\right )-2 b^4 c^2 x^2 \left (c+d x^2\right )\right )}{3 a^2 c \sqrt{\frac{b}{a}} \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]

[Out]

(Sqrt[b/a]*x*(3*a^4*d^3 + 6*a^3*b*d^3*x^2 - 2*b^4*c^2*x^2*(c + d*x^2) + a^2*b^2*
d*(8*c^2 + 8*c*d*x^2 + 3*d^2*x^4) + a*b^3*c*(-3*c^2 + 4*c*d*x^2 + 7*d^2*x^4)) +
I*b*c*(-2*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[
1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*b*c*(b^2*c
^2 - 4*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*
EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*c*(-(b*c) + a*d
)^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])

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Maple [B]  time = 0.044, size = 964, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)

[Out]

-1/3*(-3*x^5*a^2*b^2*d^3*(-b/a)^(1/2)-7*x^5*a*b^3*c*d^2*(-b/a)^(1/2)+2*x^5*b^4*c
^2*d*(-b/a)^(1/2)+6*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^2*b^2*c*d^2*
((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-8*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/
2))*x^2*a*b^3*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+2*EllipticF(x*(-b/a)
^(1/2),(a*d/b/c)^(1/2))*x^2*b^4*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+3*El
lipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^2*b^2*c*d^2*((b*x^2+a)/a)^(1/2)*((
d*x^2+c)/c)^(1/2)+7*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a*b^3*c^2*d*((
b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-2*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)
)*x^2*b^4*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-6*x^3*a^3*b*d^3*(-b/a)^(1/
2)-8*x^3*a^2*b^2*c*d^2*(-b/a)^(1/2)-4*x^3*a*b^3*c^2*d*(-b/a)^(1/2)+2*x^3*b^4*c^3
*(-b/a)^(1/2)+6*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b*c*d^2*((b*x^2+a)
/a)^(1/2)*((d*x^2+c)/c)^(1/2)-8*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^
2*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+2*EllipticF(x*(-b/a)^(1/2),(a*d/
b/c)^(1/2))*a*b^3*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+3*EllipticE(x*(-b/
a)^(1/2),(a*d/b/c)^(1/2))*a^3*b*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+7*
EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^2*c^2*d*((b*x^2+a)/a)^(1/2)*((d*
x^2+c)/c)^(1/2)-2*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^3*c^3*((b*x^2+a)
/a)^(1/2)*((d*x^2+c)/c)^(1/2)-3*x*a^4*d^3*(-b/a)^(1/2)-8*x*a^2*b^2*c^2*d*(-b/a)^
(1/2)+3*x*a*b^3*c^3*(-b/a)^(1/2))/(d*x^2+c)^(1/2)/(a*d-b*c)^3/a^2/(-b/a)^(1/2)/c
/(b*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((b^2*d*x^6 + (b^2*c + 2*a*b*d)*x^4 + a^2*c + (2*a*b*c + a^2*d)*x^2)*
sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)