3.204 \(\int \frac{\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=445 \[ \frac{b \sqrt{c} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]

[Out]

((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*Sqrt[a + b*x^2]
)/(15*c*d^3*Sqrt[c + d*x^2]) - ((b*c - a*d)*x*(a + b*x^2)^(5/2))/(c*d*Sqrt[c + d
*x^2]) - (b*(24*b^2*c^2 - 43*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])/(15*c*d^3) + (b*(6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*c*
d^2) - ((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*Sqrt[c]*d^(7/
2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*(24*b^2*c
^2 - 61*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[
c]], 1 - (b*c)/(a*d)])/(15*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c
+ d*x^2])

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Rubi [A]  time = 1.01467, antiderivative size = 445, 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 \[ \frac{b \sqrt{c} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]

[Out]

((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*Sqrt[a + b*x^2]
)/(15*c*d^3*Sqrt[c + d*x^2]) - ((b*c - a*d)*x*(a + b*x^2)^(5/2))/(c*d*Sqrt[c + d
*x^2]) - (b*(24*b^2*c^2 - 43*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])/(15*c*d^3) + (b*(6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*c*
d^2) - ((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*Sqrt[c]*d^(7/
2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*(24*b^2*c
^2 - 61*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[
c]], 1 - (b*c)/(a*d)])/(15*d^(7/2)*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 = 143.453, size = 418, normalized size = 0.94 \[ \frac{a^{\frac{3}{2}} \sqrt{b} \sqrt{c + d x^{2}} \left (45 a^{2} d^{2} - 61 a b c d + 24 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 c d^{3} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{b x \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (5 a d - 6 b c\right )}{5 c d^{2}} - \frac{b x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (15 a^{2} d^{2} - 43 a b c d + 24 b^{2} c^{2}\right )}{15 c d^{3}} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )}{c d \sqrt{c + d x^{2}}} - \frac{x \sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right )}{15 c d^{3} \sqrt{c + d x^{2}}} + \frac{\sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{15 \sqrt{c} d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(7/2)/(d*x**2+c)**(3/2),x)

[Out]

a**(3/2)*sqrt(b)*sqrt(c + d*x**2)*(45*a**2*d**2 - 61*a*b*c*d + 24*b**2*c**2)*ell
iptic_f(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(15*c*d**3*sqrt(a*(c + d*x**2)/
(c*(a + b*x**2)))*sqrt(a + b*x**2)) - b*x*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(
5*a*d - 6*b*c)/(5*c*d**2) - b*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(15*a**2*d**2
- 43*a*b*c*d + 24*b**2*c**2)/(15*c*d**3) + x*(a + b*x**2)**(5/2)*(a*d - b*c)/(c*
d*sqrt(c + d*x**2)) - x*sqrt(a + b*x**2)*(15*a**3*d**3 - 103*a**2*b*c*d**2 + 128
*a*b**2*c**2*d - 48*b**3*c**3)/(15*c*d**3*sqrt(c + d*x**2)) + sqrt(a + b*x**2)*(
15*a**3*d**3 - 103*a**2*b*c*d**2 + 128*a*b**2*c**2*d - 48*b**3*c**3)*elliptic_e(
atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(15*sqrt(c)*d**(7/2)*sqrt(c*(a + b*x**2)
/(a*(c + d*x**2)))*sqrt(c + d*x**2))

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Mathematica [C]  time = 2.09831, size = 318, normalized size = 0.71 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (15 a^3 d^3-45 a^2 b c d^2+a b^2 c d \left (61 c+16 d x^2\right )-3 b^3 c \left (8 c^2+2 c d x^2-d^2 x^4\right )\right )+4 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-15 a^3 d^3+41 a^2 b c d^2-38 a b^2 c^2 d+12 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-103 a^2 b c d^2+128 a b^2 c^2 d-48 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{15 c d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(-45*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*c*d*(61*c + 16*
d*x^2) - 3*b^3*c*(8*c^2 + 2*c*d*x^2 - d^2*x^4)) + I*b*c*(-48*b^3*c^3 + 128*a*b^2
*c^2*d - 103*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E
llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*b*c*(12*b^3*c^3 - 38*a*b^2
*c^2*d + 41*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El
lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*c*d^4*Sqrt[a + b*x^2
]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.061, size = 755, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(7/2)/(d*x^2+c)^(3/2),x)

[Out]

1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(3*(-b/a)^(1/2)*x^7*b^4*c*d^3+19*(-b/a)^(1/
2)*x^5*a*b^3*c*d^3-6*(-b/a)^(1/2)*x^5*b^4*c^2*d^2+15*(-b/a)^(1/2)*x^3*a^3*b*d^4-
29*(-b/a)^(1/2)*x^3*a^2*b^2*c*d^3+55*(-b/a)^(1/2)*x^3*a*b^3*c^2*d^2-24*(-b/a)^(1
/2)*x^3*b^4*c^3*d+60*((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*b*c*d^3-164*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*E
llipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^2*c^2*d^2+152*((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^3*c^3*d-48*((
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^4*c^4-15*((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*b*c*d^3+103*((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^2*c^2*d^2-128*((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^3*c^3*d+48*((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^4*c^4+15
*x*a^4*d^4*(-b/a)^(1/2)-45*(-b/a)^(1/2)*x*a^3*b*c*d^3+61*(-b/a)^(1/2)*x*a^2*b^2*
c^2*d^2-24*(-b/a)^(1/2)*x*a*b^3*c^3*d)/d^4/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/(-b/a)^
(1/2)/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2),x, algorithm="fricas")

[Out]

integral((b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2 + a)/(d*x^2 + c)
^(3/2), 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((b*x**2+a)**(7/2)/(d*x**2+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2), x)