3.203 \(\int \frac{1}{\left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=334 \[ \frac{4 b x \sqrt{c+d x^2} (b c-2 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2}+\frac{\sqrt{b} \sqrt{c+d x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{15 a^{5/2} \sqrt{a+b x^2} (b c-a d)^3 \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 a^3 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{c+d x^2}}{5 a \left (a+b x^2\right )^{5/2} (b c-a d)} \]

[Out]

(b*x*Sqrt[c + d*x^2])/(5*a*(b*c - a*d)*(a + b*x^2)^(5/2)) + (4*b*(b*c - 2*a*d)*x
*Sqrt[c + d*x^2])/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (Sqrt[b]*(8*b^2*c^2
- 23*a*b*c*d + 23*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)*(b*c - a*d)^3*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^
2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sqrt[d]*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*S
qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*
(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 = 0.703619, antiderivative size = 334, 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 \[ \frac{4 b x \sqrt{c+d x^2} (b c-2 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2}+\frac{\sqrt{b} \sqrt{c+d x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{15 a^{5/2} \sqrt{a+b x^2} (b c-a d)^3 \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 a^3 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{c+d x^2}}{5 a \left (a+b x^2\right )^{5/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*Sqrt[c + d*x^2])/(5*a*(b*c - a*d)*(a + b*x^2)^(5/2)) + (4*b*(b*c - 2*a*d)*x
*Sqrt[c + d*x^2])/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (Sqrt[b]*(8*b^2*c^2
- 23*a*b*c*d + 23*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)*(b*c - a*d)^3*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^
2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sqrt[d]*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*S
qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*
(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 = 120.703, size = 301, normalized size = 0.9 \[ - \frac{b x \sqrt{c + d x^{2}}}{5 a \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )} - \frac{4 b x \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{15 a^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{\sqrt{c} \sqrt{d} \sqrt{a + b x^{2}} \left (15 a^{2} d^{2} - 11 a b c d + 4 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{15 a^{3} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} - \frac{\sqrt{b} \sqrt{c + d x^{2}} \left (23 a^{2} d^{2} - 23 a b c d + 8 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 a^{\frac{5}{2}} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x*sqrt(c + d*x**2)/(5*a*(a + b*x**2)**(5/2)*(a*d - b*c)) - 4*b*x*sqrt(c + d*x
**2)*(2*a*d - b*c)/(15*a**2*(a + b*x**2)**(3/2)*(a*d - b*c)**2) + sqrt(c)*sqrt(d
)*sqrt(a + b*x**2)*(15*a**2*d**2 - 11*a*b*c*d + 4*b**2*c**2)*elliptic_f(atan(sqr
t(d)*x/sqrt(c)), 1 - b*c/(a*d))/(15*a**3*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*s
qrt(c + d*x**2)*(a*d - b*c)**3) - 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*a**
(5/2)*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 = 1.05424, size = 301, normalized size = 0.9 \[ \frac{b x \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (\left (a+b x^2\right )^2 \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right )+3 a^2 (b c-a d)^2+4 a \left (a+b x^2\right ) (b c-2 a d) (b c-a d)\right )+i \sqrt{\frac{b x^2}{a}+1} \left (a+b x^2\right )^2 \sqrt{\frac{d x^2}{c}+1} \left (b c \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+\left (15 a^3 d^3-34 a^2 b c d^2+27 a b^2 c^2 d-8 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{15 a^3 \sqrt{\frac{b}{a}} \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[b/a]*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2 + 4*a*(b*c - 2*a*d)*(b*c - a*d)*
(a + b*x^2) + (8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*(a + b*x^2)^2) + I*(a + b*x^
2)^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(b*c*(8*b^2*c^2 - 23*a*b*c*d + 23*a
^2*d^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-8*b^3*c^3 + 27*a*b^2*
c^2*d - 34*a^2*b*c*d^2 + 15*a^3*d^3)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*
c)]))/(15*a^3*Sqrt[b/a]*(b*c - a*d)^3*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])

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Maple [B]  time = 0.049, size = 1607, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(23*x^7*a*b^4*c*d^2*(-b/a)^(1/2)+54*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2
))*x^2*a^2*b^3*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+46*EllipticE(x*(-b/
a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^3*b^2*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1
/2)-46*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^2*b^3*c^2*d*((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^4*a^2*
b^3*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+27*EllipticF(x*(-b/a)^(1/2),(a
*d/b/c)^(1/2))*x^4*a*b^4*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+23*Ellipt
icE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^4*a^2*b^3*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^
2+c)/c)^(1/2)-23*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^4*a*b^4*c^2*d*((b*x
^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-20*x^3*a*b^4*c^3*(-b/a)^(1/2)-15*x*a^2*b^3*c^
3*(-b/a)^(1/2)+15*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^5*d^3*((b*x^2+a)/a
)^(1/2)*((d*x^2+c)/c)^(1/2)-23*x^7*a^2*b^3*d^3*(-b/a)^(1/2)-8*x^7*b^5*c^2*d*(-b/
a)^(1/2)-54*x^5*a^3*b^2*d^3*(-b/a)^(1/2)+16*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(
1/2))*x^2*a*b^4*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))*a^4*b*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+27*
EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b^2*c^2*d*((b*x^2+a)/a)^(1/2)*((d*
x^2+c)/c)^(1/2)+23*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^4*b*c*d^2*((b*x^2
+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-23*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^
3*b^2*c^2*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+15*EllipticF(x*(-b/a)^(1/2),
(a*d/b/c)^(1/2))*x^4*a^3*b^2*d^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+30*Elli
pticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a^4*b*d^3*((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^4*c^3*((b*x^2+a
)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-68*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*
a^3*b^2*c*d^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+35*x^5*a^2*b^3*c*d^2*(-b/a
)^(1/2)+3*x^5*a*b^4*c^2*d*(-b/a)^(1/2)-13*x^3*a^3*b^2*c*d^2*(-b/a)^(1/2)+43*x^3*
a^2*b^3*c^2*d*(-b/a)^(1/2)-34*x*a^4*b*c*d^2*(-b/a)^(1/2)-34*x^3*a^4*b*d^3*(-b/a)
^(1/2)+41*x*a^3*b^2*c^2*d*(-b/a)^(1/2)-8*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2
))*x^4*b^5*c^3*((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))*x^4*b^5*c^3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-8*Ellipti
cF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^3*c^3*((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^3*c^3*((b*x^2+a)/a)^(1/
2)*((d*x^2+c)/c)^(1/2)-8*x^5*b^5*c^3*(-b/a)^(1/2))/(d*x^2+c)^(1/2)/(a*d-b*c)^3/(
-b/a)^(1/2)/a^3/(b*x^2+a)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{2}} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

integral(1/((b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2 + a)*sqrt(d*x
^2 + c)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{2}} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)), x)