3.1.50 \(\int \frac {e+f x^2}{\sqrt {a+b x^2} (c-d x^2)^{3/2}} \, dx\) [50]

Optimal. Leaf size=237 \[ \frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}-\frac {(d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {541, 538, 438, 437, 435, 432, 430} \begin {gather*} -\frac {\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} (c f+d e) E\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2} (a d+b c)}+\frac {e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} F\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 7.38, size = 213, normalized size = 0.90 \begin {gather*} \frac {\sqrt {\frac {b}{a}} d (d e+c f) x \left (a+b x^2\right )-i b c (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d (b c+a d) \sqrt {a+b x^2} \sqrt {c-d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 333, normalized size = 1.41

method	result	size
default	\(\frac {\left (\sqrt {\frac {d}{c}}\, b c f \,x^{3}+\sqrt {\frac {d}{c}}\, b d e \,x^{3}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b c e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c f -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {d}{c}}\, a c f x +\sqrt {\frac {d}{c}}\, a d e x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}{c \sqrt {\frac {d}{c}}\, \left (a d +b c \right ) \left (-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c \right )}\)	\(333\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (-\frac {\left (-b d \,x^{2}-a d \right ) x \left (c f +d e \right )}{d c \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (-b d \,x^{2}-a d \right )}}+\frac {\left (-\frac {f}{d}+\frac {c f +d e}{d c}-\frac {a \left (c f +d e \right )}{c \left (a d +b c \right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}}+\frac {\left (c f +d e \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a d +b c \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\)	\(376\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e)/(-d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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((f*x^2+e)/(-d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e + f*x**2)/(sqrt(a + b*x**2)*(c - d*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((f*x^2+e)/(-d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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