3.1.84 \(\int \frac {(c+d x^2)^{3/2}}{(a+b x^2) (e+f x^2)^{3/2}} \, dx\) [84]

Optimal. Leaf size=223 \[ \frac {(d e-c f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {f} (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {c^{3/2} (b c-a d) \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a \sqrt {d} e (b e-a f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]

[Out]

(-c*f+d*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(
1/2))*(d*x^2+c)^(1/2)/(-a*f+b*e)/e^(1/2)/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+c^(3/2)*(-a*d
+b*c)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),1-b*c/a/d,(1-c*f/
d/e)^(1/2))*(f*x^2+e)^(1/2)/a/e/(-a*f+b*e)/d^(1/2)/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {556, 553, 422} \begin {gather*} \frac {c^{3/2} \sqrt {e+f x^2} (b c-a d) \Pi \left (1-\frac {b c}{a d};\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (b e-a f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {c+d x^2} (d e-c f) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {f} \sqrt {e+f x^2} (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*e - c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*Sqrt[f]*(b*e -
a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)*Sqrt[e + f*x^2]*EllipticPi[
1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[
(c*(e + f*x^2))/(e*(c + d*x^2))])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 553

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[c*(Sqrt[e +
f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), Ar
cTan[Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 556

Int[((e_) + (f_.)*(x_)^2)^(3/2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e -
a*f)/(b*c - a*d), Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), In
t[Sqrt[e + f*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx &=\frac {(b c-a d) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx}{b e-a f}+\frac {(d e-c f) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{b e-a f}\\ &=\frac {(d e-c f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {f} (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {c^{3/2} (b c-a d) \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a \sqrt {d} e (b e-a f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.17, size = 304, normalized size = 1.36 \begin {gather*} \frac {a b \sqrt {\frac {d}{c}} f (d e-c f) x \left (c+d x^2\right )-i a b d e (-d e+c f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a d^2 e (b e-a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i (b c-a d)^2 e f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{a b \sqrt {\frac {d}{c}} e f (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*Sqrt[d/c]*f*(d*e - c*f)*x*(c + d*x^2) - I*a*b*d*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*
EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d^2*e*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]
*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(b*c - a*d)^2*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*
EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*b*Sqrt[d/c]*e*f*(b*e - a*f)*Sqrt[c + d*x^2]*S
qrt[e + f*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(593\) vs. \(2(273)=546\).
time = 0.15, size = 594, normalized size = 2.66

method	result	size
default	\(\frac {\left (\sqrt {-\frac {d}{c}}\, a b c d \,f^{2} x^{3}-\sqrt {-\frac {d}{c}}\, a b \,d^{2} e f \,x^{3}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} d^{2} e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a b \,d^{2} e^{2}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a b c d e f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a b \,d^{2} e^{2}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{2} d^{2} e f +2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a b c d e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b^{2} c^{2} e f +\sqrt {-\frac {d}{c}}\, a b \,c^{2} f^{2} x -\sqrt {-\frac {d}{c}}\, a b c d e f x \right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{b a \sqrt {-\frac {d}{c}}\, e f \left (a f -b e \right ) \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\)	\(594\)
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+c f \right ) \left (c f -d e \right ) x}{f \left (a f -b e \right ) e \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right ) d^{2}}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, b f}-\frac {d \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right ) c}{\left (a f -b e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right ) e}{\left (a f -b e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}-\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d^{2}}{\left (a f -b e \right ) b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {2 \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c d}{\left (a f -b e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c^{2}}{\left (a f -b e \right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(724\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(3/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((-d/c)^(1/2)*a*b*c*d*f^2*x^3-(-d/c)^(1/2)*a*b*d^2*e*f*x^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))*a^2*d^2*e*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))*a*b*d^2*e^2-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a
*b*c*d*e*f+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*d^2*e^2-((d*x
^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*d^2*e*f+2*
((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b*c*d*e
*f-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c^
2*e*f+(-d/c)^(1/2)*a*b*c^2*f^2*x-(-d/c)^(1/2)*a*b*c*d*e*f*x)*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/b/a/(-d/c)^(1/2)/
e/f/(a*f-b*e)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(3/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(3/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**(3/2)/(b*x**2+a)/(f*x**2+e)**(3/2),x)

[Out]

Integral((c + d*x**2)**(3/2)/((a + b*x**2)*(e + f*x**2)**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(3/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^{3/2}}{\left (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^(3/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x)

[Out]

int((c + d*x^2)^(3/2)/((a + b*x^2)*(e + f*x^2)^(3/2)), x)

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