3.65 \(\int \frac{(c+d x^2)^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx\)

Optimal. Leaf size=400 \[ \frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{3 b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e)}{3 b^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.289189, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {543, 539, 528, 531, 418, 492, 411} \[ \frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e)}{3 b^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 543

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

Rule 539

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

Rule 528

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 531

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]

Rule 418

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

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 411

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

Rubi steps

\begin{align*} \int \frac{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx &=\frac{d \int \frac{\left (2 b c-a d+b d x^2\right ) \sqrt{e+f x^2}}{\sqrt{c+d x^2}} \, dx}{b^2}+\frac{(b c-a d)^2 \int \frac{\sqrt{e+f x^2}}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{b^2}\\ &=\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}+\frac{(b c-a d)^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{\int \frac{d (5 b c-3 a d) e+d (b d e+4 b c f-3 a d f) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^2}\\ &=\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}+\frac{(b c-a d)^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(d (5 b c-3 a d) e) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^2}+\frac{(d (b d e+4 b c f-3 a d f)) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^2}\\ &=\frac{(b d e+4 b c f-3 a d f) x \sqrt{c+d x^2}}{3 b^2 \sqrt{e+f x^2}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}+\frac{d (5 b c-3 a d) e^{3/2} \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 b^2 c \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(b c-a d)^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{(e (b d e+4 b c f-3 a d f)) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=\frac{(b d e+4 b c f-3 a d f) x \sqrt{c+d x^2}}{3 b^2 \sqrt{e+f x^2}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}-\frac{\sqrt{e} (b d e+4 b c f-3 a d 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 b^2 \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{d (5 b c-3 a d) e^{3/2} \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 b^2 c \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(b c-a d)^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^2 c \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}\\ \end{align*}

Mathematica [C]  time = 1.49034, size = 346, normalized size = 0.86 \[ \frac{-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (3 a^2 d^2 f^2-6 a b c d f^2+b^2 \left (3 c^2 f^2+c d e f-d^2 e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+f \left (a b^2 d x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right )-3 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f) \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+4 b c f+b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a b^3 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.015, size = 1059, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*((-d/c)^(1/2)*x^5*a*b^2*d^2*f^2+(-d/c)^(1/2)*x^3*a*b^2*c*d*f^2+(-d/c)^(1/2
)*x^3*a*b^2*d^2*e*f+3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a^3*d^
2*f^2-6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a^2*b*c*d*f^2+3*Elli
pticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a*b^2*c^2*f^2+EllipticF(x*(-d/c)
^(1/2),(c*f/d/e)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a*b^2*c*d*e*f-EllipticF(x*(-d/c)^(1/2),(c*f/d/
e)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a*b^2*d^2*e^2-3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*((
d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a^2*b*d^2*e*f+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*((d*x^2+c)/c)^
(1/2)*((f*x^2+e)/e)^(1/2)*a*b^2*c*d*e*f+EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)*a*b^2*d^2*e^2-3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*((d*x^2+c)/c)^(1/2)*(
(f*x^2+e)/e)^(1/2)*a^3*d^2*f^2+6*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)*a^2*b*c*d*f^2+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*((d*x^2+c
)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a^2*b*d^2*e*f-3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*((
d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a*b^2*c^2*f^2-6*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1
/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*a*b^2*c*d*e*f+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-
d/c)^(1/2))*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*b^3*c^2*e*f+(-d/c)^(1/2)*x*a*b^2*c*d*e*f)/(d*f*x^4+c*f*x^2
+d*e*x^2+c*e)/b^3/(-d/c)^(1/2)/f/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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