### 3.92 $$\int \frac{(a+c x^2)^{3/2} (d+e x+f x^2)}{g+h x} \, dx$$

Optimal. Leaf size=326 $-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 g^3 \left (f g^2-h (e g-d h)\right )\right )}{8 \sqrt{c} h^6}+\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^3}+\frac{\sqrt{a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{8 h^5}-\frac{\left (a h^2+c g^2\right )^{3/2} \left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right )}{h^6}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}$

[Out]

((8*(c*g^2 + a*h^2)*(f*g^2 - e*g*h + d*h^2) - h*(4*c*d*g*h^2 + (f*g - e*h)*(4*c*g^2 + 3*a*h^2))*x)*Sqrt[a + c*
x^2])/(8*h^5) + ((4*(f*g^2 - e*g*h + d*h^2) - 3*h*(f*g - e*h)*x)*(a + c*x^2)^(3/2))/(12*h^3) + (f*(a + c*x^2)^
(5/2))/(5*c*h) - ((3*a^2*h^4*(f*g - e*h) + 8*c^2*g^3*(f*g^2 - h*(e*g - d*h)) + 12*a*c*g*h^2*(f*g^2 - h*(e*g -
d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c]*h^6) - ((c*g^2 + a*h^2)^(3/2)*(f*g^2 - e*g*h + d*h^2)*
ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/h^6

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Rubi [A]  time = 0.76619, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.207, Rules used = {1654, 815, 844, 217, 206, 725} $-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right )}{8 \sqrt{c} h^6}+\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^3}+\frac{\sqrt{a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{8 h^5}-\frac{\left (a h^2+c g^2\right )^{3/2} \left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right )}{h^6}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((8*(c*g^2 + a*h^2)*(f*g^2 - e*g*h + d*h^2) - h*(4*c*d*g*h^2 + (f*g - e*h)*(4*c*g^2 + 3*a*h^2))*x)*Sqrt[a + c*
x^2])/(8*h^5) + ((4*(f*g^2 - e*g*h + d*h^2) - 3*h*(f*g - e*h)*x)*(a + c*x^2)^(3/2))/(12*h^3) + (f*(a + c*x^2)^
(5/2))/(5*c*h) - ((3*a^2*h^4*(f*g - e*h) + 12*a*c*g*h^2*(f*g^2 - h*(e*g - d*h)) + 8*c^2*(f*g^5 - g^3*h*(e*g -
d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c]*h^6) - ((c*g^2 + a*h^2)^(3/2)*(f*g^2 - e*g*h + d*h^2)*
ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/h^6

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx &=\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{\left (5 c d h^2-5 c h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{g+h x} \, dx}{5 c h^2}\\ &=\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{\left (5 a c^2 h^2 \left (f g^2-h (e g-4 d h)\right )-5 c^2 h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{g+h x} \, dx}{20 c^2 h^4}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{5 a c^3 h^2 \left (a h^2 \left (5 f g^2-h (5 e g-8 d h)\right )+4 c \left (f g^4-g^2 h (e g-d h)\right )\right )-5 c^3 h \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) x}{(g+h x) \sqrt{a+c x^2}} \, dx}{40 c^3 h^6}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\left (\left (c g^2+a h^2\right )^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{h^6}-\frac{\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 h^6}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac{\left (\left (c g^2+a h^2\right )^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c g^2+a h^2-x^2} \, dx,x,\frac{a h-c g x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{8 h^6}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac{\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c} h^6}-\frac{\left (c g^2+a h^2\right )^{3/2} \left (f g^2-e g h+d h^2\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{h^6}\\ \end{align*}

Mathematica [A]  time = 1.42591, size = 348, normalized size = 1.07 $\frac{\sqrt{a+c x^2} \left (3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\sqrt{c} x \left (5 a+2 c x^2\right ) \sqrt{\frac{c x^2}{a}+1}\right ) (e h-f g)}{8 \sqrt{c} h^2 \sqrt{\frac{c x^2}{a}+1}}-\frac{\left (h (d h-e g)+f g^2\right ) \left (\sqrt{\frac{c x^2}{a}+1} \left (-h \sqrt{a+c x^2} \left (8 a h^2+6 c g^2-3 c g h x+2 c h^2 x^2\right )+6 \left (a h^2+c g^2\right )^{3/2} \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right )+6 \sqrt{c} g \left (a h^2+c g^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )\right )+3 \sqrt{a} \sqrt{c} g h^2 \sqrt{a+c x^2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{6 h^6 \sqrt{\frac{c x^2}{a}+1}}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(f*(a + c*x^2)^(5/2))/(5*c*h) + ((-(f*g) + e*h)*Sqrt[a + c*x^2]*(Sqrt[c]*x*(5*a + 2*c*x^2)*Sqrt[1 + (c*x^2)/a]
+ 3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(8*Sqrt[c]*h^2*Sqrt[1 + (c*x^2)/a]) - ((f*g^2 + h*(-(e*g) + d*h))*
(3*Sqrt[a]*Sqrt[c]*g*h^2*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] + Sqrt[1 + (c*x^2)/a]*(-(h*Sqrt[a + c*x^
2]*(6*c*g^2 + 8*a*h^2 - 3*c*g*h*x + 2*c*h^2*x^2)) + 6*Sqrt[c]*g*(c*g^2 + a*h^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c
*x^2]] + 6*(c*g^2 + a*h^2)^(3/2)*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])))/(6*h^6*Sqrt[1
+ (c*x^2)/a])

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Maple [B]  time = 0.228, size = 2420, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2/h^2*c^(1/2)*g*ln((-c*g/h+(x+g/h)*c)/c^(1/2)+((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))*a*d-1/
2/h^2*c*g*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*d+3/2/h^3*c^(1/2)*g^2*ln((-c*g/h+(x+g/h)*c)/
c^(1/2)+((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))*a*e+1/h*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g
^2)/h^2)^(1/2)*a*d+3/8/h*e*a^2/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))-1/4/h^2*f*g*x*(c*x^2+a)^(3/2)+1/h^5*c^(3/
2)*g^4*ln((-c*g/h+(x+g/h)*c)/c^(1/2)+((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))*e-1/h^6*c^(3/2)*g^
5*ln((-c*g/h+(x+g/h)*c)/c^(1/2)+((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))*f-1/h^4*c^(3/2)*g^3*ln(
(-c*g/h+(x+g/h)*c)/c^(1/2)+((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))*d-2/h^3/((a*h^2+c*g^2)/h^2)^
(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+
c*g^2)/h^2)^(1/2))/(x+g/h))*a*c*g^2*d+2/h^4/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+
2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*a*c*g^3*e-2/h^5/((
a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g
/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*a*c*g^4*f+3/8/h*e*a*x*(c*x^2+a)^(1/2)+1/h^3*((x+g/h)^2*c-2*c*g/h
*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*c*g^2*d+1/h^6/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g
/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*c^2*g^5*e-3/2/
h^4*c^(1/2)*g^3*ln((-c*g/h+(x+g/h)*c)/c^(1/2)+((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))*a*f+1/2/h
^3*c*g^2*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*e-1/2/h^4*c*g^3*((x+g/h)^2*c-2*c*g/h*(x+g/h)+
(a*h^2+c*g^2)/h^2)^(1/2)*x*f-1/h^7/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2
+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*c^2*g^6*f+1/h^3*((x+g/h)^2*
c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*a*f*g^2-1/h/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g
/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*a^2*d+1
/h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*
c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*a^2*e*g-1/h^5/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^
2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g
/h))*c^2*g^4*d-1/h^2*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*a*e*g+1/h^5*((x+g/h)^2*c-2*c*g/h*(x
+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*c*g^4*f-1/h^4*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*c*g^3*e+1/3
/h*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(3/2)*d-1/h^3/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)
/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h
))*a^2*f*g^2-3/8/h^2*f*g*a*x*(c*x^2+a)^(1/2)-1/3/h^2*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(3/2)*e*g
+1/3/h^3*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(3/2)*f*g^2+1/4/h*e*x*(c*x^2+a)^(3/2)-3/8/h^2*f*g*a^2
/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+1/5*f*(c*x^2+a)^(5/2)/c/h

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22853, size = 744, normalized size = 2.28 \begin{align*} \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (\frac{4 \, c f x}{h} - \frac{5 \,{\left (c^{4} f g h^{19} - c^{4} h^{20} e\right )}}{c^{3} h^{21}}\right )} x + \frac{4 \,{\left (5 \, c^{4} f g^{2} h^{18} + 5 \, c^{4} d h^{20} + 6 \, a c^{3} f h^{20} - 5 \, c^{4} g h^{19} e\right )}}{c^{3} h^{21}}\right )} x - \frac{15 \,{\left (4 \, c^{4} f g^{3} h^{17} + 4 \, c^{4} d g h^{19} + 5 \, a c^{3} f g h^{19} - 4 \, c^{4} g^{2} h^{18} e - 5 \, a c^{3} h^{20} e\right )}}{c^{3} h^{21}}\right )} x + \frac{8 \,{\left (15 \, c^{4} f g^{4} h^{16} + 15 \, c^{4} d g^{2} h^{18} + 20 \, a c^{3} f g^{2} h^{18} + 20 \, a c^{3} d h^{20} + 3 \, a^{2} c^{2} f h^{20} - 15 \, c^{4} g^{3} h^{17} e - 20 \, a c^{3} g h^{19} e\right )}}{c^{3} h^{21}}\right )} + \frac{2 \,{\left (c^{2} f g^{6} + c^{2} d g^{4} h^{2} + 2 \, a c f g^{4} h^{2} + 2 \, a c d g^{2} h^{4} + a^{2} f g^{2} h^{4} + a^{2} d h^{6} - c^{2} g^{5} h e - 2 \, a c g^{3} h^{3} e - a^{2} g h^{5} e\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} h + \sqrt{c} g}{\sqrt{-c g^{2} - a h^{2}}}\right )}{\sqrt{-c g^{2} - a h^{2}} h^{6}} + \frac{{\left (8 \, c^{\frac{5}{2}} f g^{5} + 8 \, c^{\frac{5}{2}} d g^{3} h^{2} + 12 \, a c^{\frac{3}{2}} f g^{3} h^{2} + 12 \, a c^{\frac{3}{2}} d g h^{4} + 3 \, a^{2} \sqrt{c} f g h^{4} - 8 \, c^{\frac{5}{2}} g^{4} h e - 12 \, a c^{\frac{3}{2}} g^{2} h^{3} e - 3 \, a^{2} \sqrt{c} h^{5} e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c h^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*sqrt(c*x^2 + a)*((2*(3*(4*c*f*x/h - 5*(c^4*f*g*h^19 - c^4*h^20*e)/(c^3*h^21))*x + 4*(5*c^4*f*g^2*h^18 +
5*c^4*d*h^20 + 6*a*c^3*f*h^20 - 5*c^4*g*h^19*e)/(c^3*h^21))*x - 15*(4*c^4*f*g^3*h^17 + 4*c^4*d*g*h^19 + 5*a*c^
3*f*g*h^19 - 4*c^4*g^2*h^18*e - 5*a*c^3*h^20*e)/(c^3*h^21))*x + 8*(15*c^4*f*g^4*h^16 + 15*c^4*d*g^2*h^18 + 20*
a*c^3*f*g^2*h^18 + 20*a*c^3*d*h^20 + 3*a^2*c^2*f*h^20 - 15*c^4*g^3*h^17*e - 20*a*c^3*g*h^19*e)/(c^3*h^21)) + 2
*(c^2*f*g^6 + c^2*d*g^4*h^2 + 2*a*c*f*g^4*h^2 + 2*a*c*d*g^2*h^4 + a^2*f*g^2*h^4 + a^2*d*h^6 - c^2*g^5*h*e - 2*
a*c*g^3*h^3*e - a^2*g*h^5*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/(sqrt
(-c*g^2 - a*h^2)*h^6) + 1/8*(8*c^(5/2)*f*g^5 + 8*c^(5/2)*d*g^3*h^2 + 12*a*c^(3/2)*f*g^3*h^2 + 12*a*c^(3/2)*d*g
*h^4 + 3*a^2*sqrt(c)*f*g*h^4 - 8*c^(5/2)*g^4*h*e - 12*a*c^(3/2)*g^2*h^3*e - 3*a^2*sqrt(c)*h^5*e)*log(abs(-sqrt
(c)*x + sqrt(c*x^2 + a)))/(c*h^6)