3.168 \(\int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=309 \[ -\frac{2 c^{3/2} \sqrt{d} \sqrt{a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 a^3 \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{c+d x^2} (4 b c-3 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{15 a^{5/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d)^2 \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}} \]

[Out]

(x*Sqrt[c + d*x^2])/(5*a*(a + b*x^2)^(5/2)) + ((4*b*c - 3*a*d)*x*Sqrt[c + d*x^2]
)/(15*a^2*(b*c - a*d)*(a + b*x^2)^(3/2)) + ((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)
*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(15*a^
(5/2)*Sqrt[b]*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))
]) - (2*c^(3/2)*Sqrt[d]*(2*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d
]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(
c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.587501, antiderivative size = 309, 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 \[ -\frac{2 c^{3/2} \sqrt{d} \sqrt{a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 a^3 \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{c+d x^2} (4 b c-3 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{15 a^{5/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d)^2 \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^2]/(a + b*x^2)^(7/2),x]

[Out]

(x*Sqrt[c + d*x^2])/(5*a*(a + b*x^2)^(5/2)) + ((4*b*c - 3*a*d)*x*Sqrt[c + d*x^2]
)/(15*a^2*(b*c - a*d)*(a + b*x^2)^(3/2)) + ((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)
*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(15*a^
(5/2)*Sqrt[b]*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))
]) - (2*c^(3/2)*Sqrt[d]*(2*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d
]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*(b*c - a*d)^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 = 82.839, size = 274, normalized size = 0.89 \[ \frac{x \sqrt{c + d x^{2}}}{5 a \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \sqrt{c + d x^{2}} \left (3 a d - 4 b c\right )}{15 a^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 d \sqrt{c + d x^{2}} \left (3 a d - 2 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 a^{\frac{3}{2}} \sqrt{b} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 13 a b c d + 8 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 a^{\frac{5}{2}} \sqrt{b} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(7/2),x)

[Out]

x*sqrt(c + d*x**2)/(5*a*(a + b*x**2)**(5/2)) + x*sqrt(c + d*x**2)*(3*a*d - 4*b*c
)/(15*a**2*(a + b*x**2)**(3/2)*(a*d - b*c)) + 2*d*sqrt(c + d*x**2)*(3*a*d - 2*b*
c)*elliptic_f(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(15*a**(3/2)*sqrt(b)*sqrt
(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)*(a*d - b*c)**2) + sqrt(c + d*
x**2)*(3*a**2*d**2 - 13*a*b*c*d + 8*b**2*c**2)*elliptic_e(atan(sqrt(b)*x/sqrt(a)
), -a*d/(b*c) + 1)/(15*a**(5/2)*sqrt(b)*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sq
rt(a + b*x**2)*(a*d - b*c)**2)

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Mathematica [C]  time = 0.896881, size = 285, normalized size = 0.92 \[ \frac{x \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (\left (a+b x^2\right )^2 \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right )+3 a^2 (b c-a d)^2+a \left (a+b x^2\right ) (a d-b c) (3 a d-4 b c)\right )+i c \sqrt{\frac{b x^2}{a}+1} \left (a+b x^2\right )^2 \sqrt{\frac{d x^2}{c}+1} \left (\left (-9 a^2 d^2+17 a b c d-8 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+\left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{15 a^3 \sqrt{\frac{b}{a}} \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^(7/2),x]

[Out]

(Sqrt[b/a]*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2 + a*(-(b*c) + a*d)*(-4*b*c + 3*a*d
)*(a + b*x^2) + (8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^2) + I*c*(a + b
*x^2)^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*((8*b^2*c^2 - 13*a*b*c*d + 3*a^2
*d^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-8*b^2*c^2 + 17*a*b*c*d
- 9*a^2*d^2)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*a^3*Sqrt[b/a]*
(b*c - a*d)^2*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])

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Maple [B]  time = 0.063, size = 1411, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(1/2)/(b*x^2+a)^(7/2),x)

[Out]

1/15*(8*x^7*b^4*c^2*d*(-b/a)^(1/2)+9*x^5*a^3*b*d^3*(-b/a)^(1/2)+20*x^3*a*b^3*c^3
*(-b/a)^(1/2)-13*x^7*a*b^3*c*d^2*(-b/a)^(1/2)+8*x^5*b^4*c^3*(-b/a)^(1/2)+9*x^3*a
^4*d^3*(-b/a)^(1/2)-3*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^4*a^2*b^2*c*d^
2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+13*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^
(1/2))*x^4*a*b^3*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+18*EllipticF(x*(-
b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^3*b*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1
/2)-17*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^4*a*b^3*c^2*d*((b*x^2+a)/a)^(
1/2)*((d*x^2+c)/c)^(1/2)+3*x^7*a^2*b^2*d^3*(-b/a)^(1/2)-30*x^5*a^2*b^2*c*d^2*(-b
/a)^(1/2)+7*x^5*a*b^3*c^2*d*(-b/a)^(1/2)-17*x^3*a^3*b*c*d^2*(-b/a)^(1/2)-18*x^3*
a^2*b^2*c^2*d*(-b/a)^(1/2)-26*x*a^3*b*c^2*d*(-b/a)^(1/2)+8*EllipticF(x*(-b/a)^(1
/2),(a*d/b/c)^(1/2))*x^4*b^4*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-8*Ellip
ticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^4*b^4*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/
c)^(1/2)+9*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^4*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^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^4*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-8*EllipticE(x*(-b/a)^(1/2),
(a*d/b/c)^(1/2))*a^2*b^2*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+9*x*a^4*c*d
^2*(-b/a)^(1/2)+15*x*a^2*b^2*c^3*(-b/a)^(1/2)-16*EllipticE(x*(-b/a)^(1/2),(a*d/b
/c)^(1/2))*x^2*a*b^3*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-17*EllipticF(x*
(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2
)+13*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b*c^2*d*((b*x^2+a)/a)^(1/2)*(
(d*x^2+c)/c)^(1/2)+16*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a*b^3*c^3*((
b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-34*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2
))*x^2*a^2*b^2*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-6*EllipticE(x*(-b/a
)^(1/2),(a*d/b/c)^(1/2))*x^2*a^3*b*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)
+26*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^2*b^2*c^2*d*((b*x^2+a)/a)^(1
/2)*((d*x^2+c)/c)^(1/2)+9*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^4*a^2*b^2*
c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2))/(d*x^2+c)^(1/2)/(-b/a)^(1/2)/(a*d
-b*c)^2/a^3/(b*x^2+a)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{2} + c}}{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^2 + c)/((b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2
+ a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(7/2),x)

[Out]

Integral(sqrt(c + d*x**2)/(a + b*x**2)**(7/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(7/2), x)