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

Optimal. Leaf size=423 \[ -\frac{\sqrt{c} \sqrt{a+b x^2} (3 b c-7 a d) \left (15 a^2 d^2-11 a b c d+8 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{8 \sqrt{c} \sqrt{a+b x^2} (b c-2 a d) \left (11 a^2 d^2-11 a b c d+6 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (71 a^2 d^2-71 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{8 x \sqrt{a+b x^2} (b c-2 a d) \left (11 a^2 d^2-11 a b c d+6 b^2 c^2\right )}{105 d^3 \sqrt{c+d x^2}}-\frac{6 b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-2 a d)}{35 d^2}+\frac{b x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 d} \]

[Out]

(-8*(b*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*x*Sqrt[a + b*x^2])/(105*
d^3*Sqrt[c + d*x^2]) + (b*(24*b^2*c^2 - 71*a*b*c*d + 71*a^2*d^2)*x*Sqrt[a + b*x^
2]*Sqrt[c + d*x^2])/(105*d^3) - (6*b*(b*c - 2*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c +
d*x^2])/(35*d^2) + (b*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*d) + (8*Sqrt[c]*(b
*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcT
an[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*
(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(3*b*c - 7*a*d)*(8*b^2*c^2 - 11*a*b*c*
d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)
/(a*d)])/(105*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 = 1.06507, antiderivative size = 423, 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 \[ -\frac{\sqrt{c} \sqrt{a+b x^2} (3 b c-7 a d) \left (15 a^2 d^2-11 a b c d+8 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{8 \sqrt{c} \sqrt{a+b x^2} (b c-2 a d) \left (11 a^2 d^2-11 a b c d+6 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (71 a^2 d^2-71 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{8 x \sqrt{a+b x^2} (b c-2 a d) \left (11 a^2 d^2-11 a b c d+6 b^2 c^2\right )}{105 d^3 \sqrt{c+d x^2}}-\frac{6 b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-2 a d)}{35 d^2}+\frac{b x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 d} \]

Antiderivative was successfully verified.

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

[Out]

(-8*(b*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*x*Sqrt[a + b*x^2])/(105*
d^3*Sqrt[c + d*x^2]) + (b*(24*b^2*c^2 - 71*a*b*c*d + 71*a^2*d^2)*x*Sqrt[a + b*x^
2]*Sqrt[c + d*x^2])/(105*d^3) - (6*b*(b*c - 2*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c +
d*x^2])/(35*d^2) + (b*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*d) + (8*Sqrt[c]*(b
*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcT
an[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*
(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(3*b*c - 7*a*d)*(8*b^2*c^2 - 11*a*b*c*
d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)
/(a*d)])/(105*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 = 130.858, size = 408, normalized size = 0.96 \[ - \frac{8 \sqrt{a} \sqrt{b} \sqrt{c + d x^{2}} \left (2 a d - b c\right ) \left (11 a^{2} d^{2} - 11 a b c d + 6 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{105 d^{4} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{b x \left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}{7 d} + \frac{6 b x \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{35 d^{2}} + \frac{b x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (71 a^{2} d^{2} - 71 a b c d + 24 b^{2} c^{2}\right )}{105 d^{3}} + \frac{8 b x \sqrt{c + d x^{2}} \left (2 a d - b c\right ) \left (11 a^{2} d^{2} - 11 a b c d + 6 b^{2} c^{2}\right )}{105 d^{4} \sqrt{a + b x^{2}}} + \frac{\sqrt{c} \sqrt{a + b x^{2}} \left (7 a d - 3 b c\right ) \left (15 a^{2} d^{2} - 11 a b c d + 8 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{105 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*sqrt(a)*sqrt(b)*sqrt(c + d*x**2)*(2*a*d - b*c)*(11*a**2*d**2 - 11*a*b*c*d + 6
*b**2*c**2)*elliptic_e(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(105*d**4*sqrt(a
*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)) + b*x*(a + b*x**2)**(5/2)*sqrt
(c + d*x**2)/(7*d) + 6*b*x*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(2*a*d - b*c)/(3
5*d**2) + b*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(71*a**2*d**2 - 71*a*b*c*d + 24*
b**2*c**2)/(105*d**3) + 8*b*x*sqrt(c + d*x**2)*(2*a*d - b*c)*(11*a**2*d**2 - 11*
a*b*c*d + 6*b**2*c**2)/(105*d**4*sqrt(a + b*x**2)) + sqrt(c)*sqrt(a + b*x**2)*(7
*a*d - 3*b*c)*(15*a**2*d**2 - 11*a*b*c*d + 8*b**2*c**2)*elliptic_f(atan(sqrt(d)*
x/sqrt(c)), 1 - b*c/(a*d))/(105*d**(7/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*s
qrt(c + d*x**2))

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Mathematica [C]  time = 2.72893, size = 321, normalized size = 0.76 \[ \frac{b d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (122 a^2 d^2+a b d \left (66 d x^2-89 c\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )-8 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (22 a^3 d^3-33 a^2 b c d^2+23 a b^2 c^2 d-6 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (105 a^4 d^4-298 a^3 b c d^3+353 a^2 b^2 c^2 d^2-208 a b^3 c^3 d+48 b^4 c^4\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{105 d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(122*a^2*d^2 + a*b*d*(-89*c + 66*d*x^2)
+ 3*b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) - (8*I)*b*c*(-6*b^3*c^3 + 23*a*b^2*c^2
*d - 33*a^2*b*c*d^2 + 22*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipt
icE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(48*b^4*c^4 - 208*a*b^3*c^3*d + 353
*a^2*b^2*c^2*d^2 - 298*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (
d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*Sqrt[b/a]*d^4*Sqr
t[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.033, size = 852, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(15*(-b/a)^(1/2)*x^9*b^4*d^4+81*(-b/a)^(1/
2)*x^7*a*b^3*d^4-3*(-b/a)^(1/2)*x^7*b^4*c*d^3+188*(-b/a)^(1/2)*x^5*a^2*b^2*d^4-2
6*(-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+122*(-b/a)^(1/2)*x
^3*a^3*b*d^4+99*(-b/a)^(1/2)*x^3*a^2*b^2*c*d^3-83*(-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+105*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellip
ticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^4*d^4-298*((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+353*((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^2*c^2
*d^2-208*((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+176*((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^3-264*((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+184*
((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+122*(-b/a)^(1/2)*x*a^3*b*c*d^3-89*(-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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((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. \[{\rm integral}\left (\frac{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/2)/sqrt(d*x^2 + c),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)/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((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. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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