3.28 \(\int \frac{\left (a+b x^2\right ) \sqrt{e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx\)
Optimal. Leaf size=385 \[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 d \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left (c+d x^2\right )^{3/2} (d e-c f)}+\frac{\sqrt{e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{3/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]
[Out]
-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*d*(c + d*x^2)^(5/2)) + ((a*d*(4*d*e - 3*c*
f) + b*c*(d*e - 2*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*d*(d*e - c*f)*(c + d*x^2)^(3/
2)) + ((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^
2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])
/(15*c^(5/2)*d^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c +
d*x^2))]) - (e^(3/2)*Sqrt[f]*(2*a*d*(2*d*e - 3*c*f) + b*c*(d*e + c*f))*Sqrt[c +
d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*d*(d*e -
c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
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Rubi [A] time = 1.12132, antiderivative size = 385, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 d \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left (c+d x^2\right )^{3/2} (d e-c f)}+\frac{\sqrt{e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{3/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]
[Out]
-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*d*(c + d*x^2)^(5/2)) + ((a*d*(4*d*e - 3*c*
f) + b*c*(d*e - 2*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*d*(d*e - c*f)*(c + d*x^2)^(3/
2)) + ((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^
2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])
/(15*c^(5/2)*d^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c +
d*x^2))]) - (e^(3/2)*Sqrt[f]*(2*a*d*(2*d*e - 3*c*f) + b*c*(d*e + c*f))*Sqrt[c +
d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*d*(d*e -
c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(f*x**2+e)**(1/2)/(d*x**2+c)**(7/2),x)
[Out]
Timed out
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Mathematica [C] time = 2.30643, size = 379, normalized size = 0.98 \[ \frac{-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (-\left (c+d x^2\right )^2 \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right )+3 c^2 (b c-a d) (d e-c f)^2-c \left (c+d x^2\right ) (d e-c f) (a d (4 d e-3 c f)+b c (d e-2 c f))\right )+i e \left (c+d x^2\right )^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (\left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-(c f-d e) (a d (9 c f-8 d e)+b c (c f-2 d e)) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 c^4 \left (\frac{d}{c}\right )^{3/2} \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (d e-c f)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]
[Out]
(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e - c*f)*(a*d
*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*(c + d*x^2) - (2*b*c*(d^2*e^2 - c*d*e*f +
c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*(c + d*x^2)^2)) + I*e*(c +
d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((2*b*c*(d^2*e^2 - c*d*e*f + c^
2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]
*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(b*c*(-2*d*e + c*f) + a*d*(-8*d*e + 9*c*f))*E
llipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*c^4*(d/c)^(3/2)*(d*e - c*f)^
2*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])
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Maple [B] time = 0.08, size = 2856, normalized size = 7.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
[Out]
1/15*(x^3*b*c^5*f^3*(-d/c)^(1/2)+2*x^5*b*c*d^4*e^3*(-d/c)^(1/2)+9*x^3*a*c^4*d*f^
3*(-d/c)^(1/2)+20*x^3*a*c*d^4*e^3*(-d/c)^(1/2)+5*x^3*b*c^2*d^3*e^3*(-d/c)^(1/2)+
15*x*a*c^2*d^3*e^3*(-d/c)^(1/2)+x*b*c^5*e*f^2*(-d/c)^(1/2)+3*x^7*a*c^2*d^3*f^3*(
-d/c)^(1/2)-13*x^7*a*c*d^4*e*f^2*(-d/c)^(1/2)-2*x^7*b*c^2*d^3*e*f^2*(-d/c)^(1/2)
+2*x^7*b*c*d^4*e^2*f*(-d/c)^(1/2)-30*x^5*a*c^2*d^3*e*f^2*(-d/c)^(1/2)+7*x^5*a*c*
d^4*e^2*f*(-d/c)^(1/2)-5*x^5*b*c^3*d^2*e*f^2*(-d/c)^(1/2)+3*x^5*b*c^2*d^3*e^2*f*
(-d/c)^(1/2)-17*x^3*a*c^3*d^2*e*f^2*(-d/c)^(1/2)-18*x^3*a*c^2*d^3*e^2*f*(-d/c)^(
1/2)+7*x^3*b*c^4*d*e*f^2*(-d/c)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*x^4*b*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*x^3*b*c^3*d^2*e^2*f*
(-d/c)^(1/2)+9*x*a*c^4*d*e*f^2*(-d/c)^(1/2)-26*x*a*c^3*d^2*e^2*f*(-d/c)^(1/2)+x*
b*c^4*d*e^2*f*(-d/c)^(1/2)+8*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*d^5
*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticE(x*(-d/c)^(1/2),(c*f/d/e
)^(1/2))*x^4*a*d^5*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticF(x*(-d
/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)
+EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^5*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x
^2+e)/e)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*d^2*e^3*((d*x^2
+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c
^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c
*f/d/e)^(1/2))*b*c^5*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticE(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*d^2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(
1/2)+8*x^7*a*d^5*e^2*f*(-d/c)^(1/2)+2*x^7*b*c^3*d^2*f^3*(-d/c)^(1/2)+9*x^5*a*c^3
*d^2*f^3*(-d/c)^(1/2)+6*x^5*b*c^4*d*f^3*(-d/c)^(1/2)-3*EllipticE(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+13*
EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*e^2*f*((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c^3*d^2*e*
f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)
^(1/2))*x^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+18*EllipticF
(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2
+e)/e)^(1/2)-6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^3*d^2*e*f^2*((d
*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+26*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*x^2*a*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-4*EllipticE(x*(-d/
c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1
/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((d*x^2+c)/c
)^(1/2)*((f*x^2+e)/e)^(1/2)-34*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c
^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*x^2*b*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*Ell
ipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x
^2+e)/e)^(1/2)+16*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*
x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-16*EllipticE(x*(-d/c)^
(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-4
*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)
*((f*x^2+e)/e)^(1/2)+9*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^4*d*e*f^2*(
(d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-17*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/
2))*a*c^3*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticF(x*(-d/c)
^(1/2),(c*f/d/e)^(1/2))*b*c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*
EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*
x^2+e)/e)^(1/2)+13*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*d^2*e^2*f*((d
*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))
*b*c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticF(x*(-d/c)^(1/2
),(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+9
*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/
2)*((f*x^2+e)/e)^(1/2)-17*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*
e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticF(x*(-d/c)^(1/2),(c*f/d/e)
^(1/2))*x^4*b*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticF(
x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)+8*x^5*a*d^5*e^3*(-d/c)^(1/2))/(f*x^2+e)^(1/2)/(-d/c)^(1/2)/(c*f-d*e)
^2/c^3/d/(d*x^2+c)^(5/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2),x, algorithm="maxima")
[Out]
integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}{{\left (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )} \sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2),x, algorithm="fricas")
[Out]
integral((b*x^2 + a)*sqrt(f*x^2 + e)/((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3
)*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)*(f*x**2+e)**(1/2)/(d*x**2+c)**(7/2),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2),x, algorithm="giac")
[Out]
integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)