### 3.96 $$\int \frac{(a+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^5} \, dx$$

Optimal. Leaf size=511 $\frac{\left (a+c x^2\right )^{3/2} \left (-3 h x \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 g^2 \left (5 f g^2-h (d h+e g)\right )\right )+4 a^2 h^4 (f g-2 e h)-a c g h^2 \left (25 f g^2-h (5 e g-9 d h)\right )-4 c^2 g^4 (5 f g-e h)\right )}{24 h^3 (g+h x)^3 \left (a h^2+c g^2\right )^2}+\frac{c \sqrt{a+c x^2} \left (h x \left (12 a^2 f h^4+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )+4 c^2 g^3 (5 f g-e h)\right )+8 \left (a h^2+c g^2\right )^2 (5 f g-e h)\right )}{8 h^5 (g+h x) \left (a h^2+c g^2\right )^2}-\frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{8 h^6 \left (a h^2+c g^2\right )^{5/2}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (5 f g-e h)}{h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}$

[Out]

(c*(8*(5*f*g - e*h)*(c*g^2 + a*h^2)^2 + h*(12*a^2*f*h^4 + 4*c^2*g^3*(5*f*g - e*h) + a*c*h^2*(35*f*g^2 - h*(7*e
*g - 3*d*h)))*x)*Sqrt[a + c*x^2])/(8*h^5*(c*g^2 + a*h^2)^2*(g + h*x)) + ((4*a^2*h^4*(f*g - 2*e*h) - 4*c^2*g^4*
(5*f*g - e*h) - a*c*g*h^2*(25*f*g^2 - h*(5*e*g - 9*d*h)) - 3*h*(4*a^2*f*h^4 + a*c*h^2*(17*f*g^2 - h*(5*e*g - d
*h)) + 2*c^2*g^2*(5*f*g^2 - h*(e*g + d*h)))*x)*(a + c*x^2)^(3/2))/(24*h^3*(c*g^2 + a*h^2)^2*(g + h*x)^3) - ((f
*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(4*h*(c*g^2 + a*h^2)*(g + h*x)^4) - (c^(3/2)*(5*f*g - e*h)*ArcTanh[(S
qrt[c]*x)/Sqrt[a + c*x^2]])/h^6 - (c*(12*a^3*f*h^6 + 8*c^3*g^5*(5*f*g - e*h) + 20*a*c^2*g^3*h^2*(5*f*g - e*h)
+ 3*a^2*c*h^4*(25*f*g^2 - h*(5*e*g - d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(8*h
^6*(c*g^2 + a*h^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.0919, antiderivative size = 511, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.241, Rules used = {1651, 811, 813, 844, 217, 206, 725} $\frac{\left (a+c x^2\right )^{3/2} \left (-3 x \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (d h+e g)\right )\right )+4 a^2 h^3 (f g-2 e h)-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-\frac{4 c^2 g^4 (5 f g-e h)}{h}\right )}{24 h^2 (g+h x)^3 \left (a h^2+c g^2\right )^2}+\frac{c \sqrt{a+c x^2} \left (h x \left (12 a^2 f h^4+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )+4 c^2 g^3 (5 f g-e h)\right )+8 \left (a h^2+c g^2\right )^2 (5 f g-e h)\right )}{8 h^5 (g+h x) \left (a h^2+c g^2\right )^2}-\frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{8 h^6 \left (a h^2+c g^2\right )^{5/2}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (5 f g-e h)}{h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^5,x]

[Out]

(c*(8*(5*f*g - e*h)*(c*g^2 + a*h^2)^2 + h*(12*a^2*f*h^4 + 4*c^2*g^3*(5*f*g - e*h) + a*c*h^2*(35*f*g^2 - h*(7*e
*g - 3*d*h)))*x)*Sqrt[a + c*x^2])/(8*h^5*(c*g^2 + a*h^2)^2*(g + h*x)) + ((4*a^2*h^3*(f*g - 2*e*h) - (4*c^2*g^4
*(5*f*g - e*h))/h - a*c*g*h*(25*f*g^2 - h*(5*e*g - 9*d*h)) - 3*(4*a^2*f*h^4 + a*c*h^2*(17*f*g^2 - h*(5*e*g - d
*h)) + 2*c^2*(5*f*g^4 - g^2*h*(e*g + d*h)))*x)*(a + c*x^2)^(3/2))/(24*h^2*(c*g^2 + a*h^2)^2*(g + h*x)^3) - ((f
*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(4*h*(c*g^2 + a*h^2)*(g + h*x)^4) - (c^(3/2)*(5*f*g - e*h)*ArcTanh[(S
qrt[c]*x)/Sqrt[a + c*x^2]])/h^6 - (c*(12*a^3*f*h^6 + 8*c^3*g^5*(5*f*g - e*h) + 20*a*c^2*g^3*h^2*(5*f*g - e*h)
+ 3*a^2*c*h^4*(25*f*g^2 - h*(5*e*g - d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(8*h
^6*(c*g^2 + a*h^2)^(5/2))

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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)^5} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\int \frac{\left (-4 (c d g-a f g+a e h)-\left (4 a f h-c \left (e g-\frac{5 f g^2}{h}-d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^4} \, dx}{4 \left (c g^2+a h^2\right )}\\ &=\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac{\int \frac{\left (-4 a c \left (5 c f g^3-c g h (e g+3 d h)+4 a h^2 (2 f g-e h)\right )+\frac{2 c \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x}{h}\right ) \sqrt{a+c x^2}}{(g+h x)^2} \, dx}{16 h^2 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\int \frac{-4 a c \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right )+\frac{32 c^2 (5 f g-e h) \left (c g^2+a h^2\right )^2 x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{32 h^4 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\left (c^2 (5 f g-e h)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{h^6}+\frac{\left (c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{8 h^6 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\left (c^2 (5 f g-e h)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{\left (c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\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 )}{8 h^6 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{c^{3/2} (5 f g-e h) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{8 h^6 \left (c g^2+a h^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 2.52245, size = 575, normalized size = 1.13 $-\frac{\frac{h \sqrt{a+c x^2} \left (-c (g+h x)^3 \left (4 a^2 h^4 (31 f g-8 e h)+a c g h^2 \left (h (15 d h-91 e g)+287 f g^2\right )+2 c^2 \left (g^3 h (3 d h-25 e g)+77 f g^5\right )\right )+(g+h x)^2 \left (a h^2+c g^2\right ) \left (12 a^2 f h^4+a c h^2 \left (h (15 d h-43 e g)+95 f g^2\right )+2 c^2 \left (g^2 h (9 d h-23 e g)+43 f g^4\right )\right )-2 (g+h x) \left (a h^2+c g^2\right )^2 \left (-4 a h^2 (e h-2 f g)+c g h (9 d h-13 e g)+17 c f g^3\right )+6 \left (a h^2+c g^2\right )^3 \left (h (d h-e g)+f g^2\right )-24 c f (g+h x)^4 \left (a h^2+c g^2\right )^2\right )}{(g+h x)^4 \left (a h^2+c g^2\right )^2}+\frac{3 c \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (3 a^2 c h^4 \left (h (d h-5 e g)+25 f g^2\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{\left (a h^2+c g^2\right )^{5/2}}-\frac{3 c \log (g+h x) \left (3 a^2 c h^4 \left (h (d h-5 e g)+25 f g^2\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{\left (a h^2+c g^2\right )^{5/2}}+24 c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) (5 f g-e h)}{24 h^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^5,x]

[Out]

-((h*Sqrt[a + c*x^2]*(6*(c*g^2 + a*h^2)^3*(f*g^2 + h*(-(e*g) + d*h)) - 2*(c*g^2 + a*h^2)^2*(17*c*f*g^3 + c*g*h
*(-13*e*g + 9*d*h) - 4*a*h^2*(-2*f*g + e*h))*(g + h*x) + (c*g^2 + a*h^2)*(12*a^2*f*h^4 + 2*c^2*(43*f*g^4 + g^2
*h*(-23*e*g + 9*d*h)) + a*c*h^2*(95*f*g^2 + h*(-43*e*g + 15*d*h)))*(g + h*x)^2 - c*(4*a^2*h^4*(31*f*g - 8*e*h)
+ 2*c^2*(77*f*g^5 + g^3*h*(-25*e*g + 3*d*h)) + a*c*g*h^2*(287*f*g^2 + h*(-91*e*g + 15*d*h)))*(g + h*x)^3 - 24
*c*f*(c*g^2 + a*h^2)^2*(g + h*x)^4))/((c*g^2 + a*h^2)^2*(g + h*x)^4) - (3*c*(12*a^3*f*h^6 + 8*c^3*g^5*(5*f*g -
e*h) + 20*a*c^2*g^3*h^2*(5*f*g - e*h) + 3*a^2*c*h^4*(25*f*g^2 + h*(-5*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h
^2)^(5/2) + 24*c^(3/2)*(5*f*g - e*h)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c*(12*a^3*f*h^6 + 8*c^3*g^5*(5*f*
g - e*h) + 20*a*c^2*g^3*h^2*(5*f*g - e*h) + 3*a^2*c*h^4*(25*f*g^2 + h*(-5*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[
c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(5/2))/(24*h^6)

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Maple [B]  time = 0.254, size = 12481, normalized size = 24.4 \begin{align*} \text{output 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)^5,x)

[Out]

result too large to display

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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)^5,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)^5,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 )}{\left (g + h x\right )^{5}}\, 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)**5,x)

[Out]

Integral((a + c*x**2)**(3/2)*(d + e*x + f*x**2)/(g + h*x)**5, x)

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Giac [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)^5,x, algorithm="giac")

[Out]

Timed out