3.1.43 \(\int \frac {(a+b x^2) (c+d x^2)^{3/2}}{(e+f x^2)^{3/2}} \, dx\) [43]

Optimal. Leaf size=358 \[ -\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) x \sqrt {c+d x^2}}{3 e f^2 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}+\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (4 b d e-3 b c f-3 a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-(-a*f+b*e)*x*(d*x^2+c)^(3/2)/e/f/(f*x^2+e)^(1/2)-1/3*(b*e*(-7*c*f+8*d*e)-3*a*f*(-c*f+2*d*e))*x*(d*x^2+c)^(1/2
)/e/f^2/(f*x^2+e)^(1/2)+1/3*(b*e*(-7*c*f+8*d*e)-3*a*f*(-c*f+2*d*e))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*El
lipticE(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)/f^(5/2)/e^(1/2)/(e*(d*x^2+c)/c/
(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/3*(-3*a*d*f-3*b*c*f+4*b*d*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*Ellipt
icF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*(d*x^2+c)^(1/2)/f^(5/2)/(e*(d*x^2+c)/c/(f*x
^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d*(-3*a*f+4*b*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/e/f^2

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Rubi [A]
time = 0.25, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {540, 542, 545, 429, 506, 422} \begin {gather*} -\frac {\sqrt {e} \sqrt {c+d x^2} (-3 a d f-3 b c f+4 b d e) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (4 b e-3 a f)}{3 e f^2}-\frac {x \sqrt {c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f))}{3 e f^2 \sqrt {e+f x^2}}-\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{e f \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*x*Sqrt[c + d*x^2])/(e*f^2*Sqrt[e + f*x^2]) - ((b*e - a*f)*x*
(c + d*x^2)^(3/2))/(e*f*Sqrt[e + f*x^2]) + (d*(4*b*e - 3*a*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f^2) + (
(b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(
c*f)])/(3*Sqrt[e]*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(4*b*d*e - 3*b*c*f
- 3*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(5/2)*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 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 540

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/(a*b*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)), Int[(a + b*x
^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*(p + 1) + (b*e - a*f)*(n*q + 1))
*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]

Rule 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 545

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{3/2}} \, dx &=-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}-\frac {\int \frac {\sqrt {c+d x^2} \left (-b c e-d (4 b e-3 a f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{e f}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}-\frac {\int \frac {c e (4 b d e-3 b c f-3 a d f)+d (b e (8 d e-7 c f)-3 a f (2 d e-c f)) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f^2}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}-\frac {(c (4 b d e-3 b c f-3 a d f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 f^2}-\frac {(d (b e (8 d e-7 c f)-3 a f (2 d e-c f))) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f^2}\\ &=-\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) x \sqrt {c+d x^2}}{3 e f^2 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}-\frac {\sqrt {e} (4 b d e-3 b c f-3 a d 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 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 f^2}\\ &=-\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) x \sqrt {c+d x^2}}{3 e f^2 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}+\frac {(b e (8 d e-7 c f)-3 a f (2 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 )}{3 \sqrt {e} f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (4 b d e-3 b c f-3 a d 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 f^{5/2} \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 = 5.21, size = 260, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (3 a f (-d e+c f)+b e \left (4 d e-3 c f+d f x^2\right )\right )-i d e (-3 a f (-2 d e+c f)+b e (-8 d e+7 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 e (-d e+c f) (-8 b d e+3 b c f+6 a d 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 )}{3 \sqrt {\frac {d}{c}} e f^3 \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(3*a*f*(-(d*e) + c*f) + b*e*(4*d*e - 3*c*f + d*f*x^2)) - I*d*e*(-3*a*f*(-2*d*e + c*
f) + b*e*(-8*d*e + 7*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*e*(-(d*e) + c*f)*(-8*b*d*e + 3*b*c*f + 6*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*Ar
cSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*Sqrt[d/c]*e*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]
time = 0.15, size = 750, normalized size = 2.09 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*((-d/c)^(1/2)*b*d^2*e*f^2*x^5+3*(-d/c)^(1/2)*a*c*d*f^3*x^3-3*(-d/c)^(1/2)*
a*d^2*e*f^2*x^3-2*(-d/c)^(1/2)*b*c*d*e*f^2*x^3+4*(-d/c)^(1/2)*b*d^2*e^2*f*x^3+6*((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*e*f^2-6*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*d^2*e^2*f+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^2*e*f^2-11*((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*e^2*f+8*((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*d
^2*e^3-3*((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*e*f^2+6*((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^2*e^2*f+7*((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*e^2*f-8*((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*d^2*e^3+3*(-d/c)^(1/2)*a*c^2*f^3*x-3*(-d/c)^(1/2)*a*c*d*e*f
^2*x-3*(-d/c)^(1/2)*b*c^2*e*f^2*x+4*(-d/c)^(1/2)*b*c*d*e^2*f*x)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/f^3/e/(-d/c)^(1/
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)^(3/2)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(d*x^2 + c)^(3/2)/(f*x^2 + e)^(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((b*x^2+a)*(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),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 {\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((b*x^2 + a)*(d*x^2 + c)^(3/2)/(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 (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (f\,x^2+e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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