3.1.40 \(\int \frac {a+b x^2}{(c+d x^2)^{5/2} \sqrt {e+f x^2}} \, dx\) [40]

Optimal. Leaf size=284 \[ -\frac {(b c-a d) x \sqrt {e+f x^2}}{3 c (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {(2 a d (d e-2 c f)+b c (d e+c f)) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} \sqrt {d} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} (2 b c e+a d e-3 a c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-1/3*(-3*a*c*f+a*d*e+2*b*c*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^
(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*f^(1/2)*(d*x^2+c)^(1/2)/c^2/(-c*f+d*e)^2/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x
^2+e)^(1/2)-1/3*(-a*d+b*c)*x*(f*x^2+e)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/3*(2*a*d*(-2*c*f+d*e)+b*c*(c*f+d*e
))*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*(f
*x^2+e)^(1/2)/c^(3/2)/(-c*f+d*e)^2/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 = 0.14, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {541, 539, 429, 422} \begin {gather*} \frac {\sqrt {e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} \sqrt {d} \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 {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (-3 a c f+a d e+2 b c e) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 \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} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]

[Out]

-1/3*((b*c - a*d)*x*Sqrt[e + f*x^2])/(c*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*a*d*(d*e - 2*c*f) + b*c*(d*e + c*
f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*Sqrt[d]*(d*e - c*f)^2*
Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt[f]*(2*b*c*e + a*d*e - 3*a*c*f)*Sqrt[c +
d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*
(e + f*x^2))]*Sqrt[e + f*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 429

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

Rule 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[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 {a+b x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{3 c (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {-b c e-2 a d e+3 a c f+(b c-a d) f x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{3 c (d e-c f)}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{3 c (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(f (2 b c e+a d e-3 a c f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c (d e-c f)^2}+\frac {(2 a d (d e-2 c f)+b c (d e+c f)) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c (d e-c f)^2}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{3 c (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {(2 a d (d e-2 c f)+b c (d e+c f)) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} \sqrt {d} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} (2 b c e+a d e-3 a c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 9.83, size = 302, normalized size = 1.06 \begin {gather*} \frac {\sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (b c \left (2 c^2 f+d^2 e x^2+c d f x^2\right )+a d \left (-5 c^2 f+2 d^2 e x^2+c d \left (3 e-4 f x^2\right )\right )\right )+i e (2 a d (d e-2 c f)+b c (d e+c f)) \left (c+d x^2\right ) \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 (-d e+c f) (b c e+2 a d e-3 a c f) \left (c+d x^2\right ) \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 )}{3 c^2 \sqrt {\frac {d}{c}} (d e-c f)^2 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]

[Out]

(Sqrt[d/c]*x*(e + f*x^2)*(b*c*(2*c^2*f + d^2*e*x^2 + c*d*f*x^2) + a*d*(-5*c^2*f + 2*d^2*e*x^2 + c*d*(3*e - 4*f
*x^2))) + I*e*(2*a*d*(d*e - 2*c*f) + b*c*(d*e + c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elli
pticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(-(d*e) + c*f)*(b*c*e + 2*a*d*e - 3*a*c*f)*(c + d*x^2)*Sqrt[1 +
(d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*c^2*Sqrt[d/c]*(d*e - c*f)^2
*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1351\) vs. \(2(324)=648\).
time = 0.12, size = 1352, normalized size = 4.76

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {x \left (a d -b c \right ) \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d^{2} c \left (c f -d e \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {\left (d f \,x^{2}+d e \right ) x \left (4 a c d f -2 a \,d^{2} e -b \,c^{2} f -b c d e \right )}{3 d \,c^{2} \left (c f -d e \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (-\frac {\left (a d -b c \right ) f}{3 d c \left (c f -d e \right )}+\frac {4 a c d f -2 a \,d^{2} e -b \,c^{2} f -b c d e}{3 \left (c f -d e \right ) d \,c^{2}}+\frac {e \left (4 a c d f -2 a \,d^{2} e -b \,c^{2} f -b c d e \right )}{3 c^{2} \left (c f -d e \right )^{2}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (4 a c d f -2 a \,d^{2} e -b \,c^{2} f -b c d e \right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{3 c^{2} \left (c f -d e \right )^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(540\)
default	\(\text {Expression too large to display}\)	\(1352\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-5*(-d/c)^(1/2)*a*c^2*d*e*f*x+2*((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*d^3*e^2*x^2-2*((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*d
^3*e^2*x^2+2*(-d/c)^(1/2)*b*c^3*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*c^3*f^2+(-d/c)^(1/2)*b*c*d^2*e*f*x^5-(-d/c)^(1/2)*a*c*d^2*e*f*x^3+(-d/c)^(1/2)*b*c^2*d*e*f*x^3+(
-d/c)^(1/2)*b*c^2*d*f^2*x^5-5*(-d/c)^(1/2)*a*c^2*d*f^2*x^3+(-d/c)^(1/2)*b*c*d^2*e^2*x^3+3*(-d/c)^(1/2)*a*c*d^2
*e^2*x+((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*d^2*e^2*x^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))*b*c*d^2*e^2*x^2-5*((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*c^2*d*e*f+4*((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*c^2*d*e*f+3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ell
ipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d*f^2*x^2+2*((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*c*d^2*e^2-((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^3*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))*b*c^2
*d*e^2-2*((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*c*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))*b*c^3*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))*b*c^2*d*e^2+2*(-d/c)^(1/2)*a*d^3*e^2*x^3+2*(-d/c)^(1/
2)*b*c^3*f^2*x^3-4*(-d/c)^(1/2)*a*c*d^2*f^2*x^5+2*(-d/c)^(1/2)*a*d^3*e*f*x^5-5*((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*c*d^2*e*f*x^2-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ell
ipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^2*d*e*f*x^2+4*((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*c*d^2*e*f*x^2-((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^2*d*e*f*x^2)/(f*x^2+e)^(1/2)/(c*f-d*e)^2/c^2/(-d/c)^(1/2)/(d*x^2+c)^(3/2)

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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((b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), 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((b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(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 {a + b x^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}} \sqrt {e + f x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral((a + b*x**2)/((c + d*x**2)**(5/2)*sqrt(e + f*x**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((b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {b\,x^2+a}{{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(1/2)),x)

[Out]

int((a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(1/2)), x)

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