3.97 \(\int \frac{(a+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^6} \, dx\)

Optimal. Leaf size=507 \[ -\frac{\left (a+c x^2\right )^{3/2} \left (h x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (3 d h^2+5 f g^2\right )+4 c^2 f g^5\right )}{12 h^3 (g+h x)^4 \left (a h^2+c g^2\right )^2}-\frac{c \sqrt{a+c x^2} \left (h x \left (a^2 c g h^4 (34 f g-3 e h)+8 a^3 f h^6+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )-a^3 h^6 (2 f g-3 e h)+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{8 h^5 (g+h x)^2 \left (a h^2+c g^2\right )^3}+\frac{c^2 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+3 a^3 h^6 (6 f g-e h)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{8 h^6 \left (a h^2+c g^2\right )^{7/2}}+\frac{c^{3/2} f \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )} \]

[Out]

-(c*(8*c^3*f*g^7 + 20*a*c^2*f*g^5*h^2 - a^3*h^6*(2*f*g - 3*e*h) + a^2*c*g*h^4*(13*f*g^2 + 3*d*h^2) + h*(12*c^3
*f*g^6 + 8*a^3*f*h^6 + a^2*c*g*h^4*(34*f*g - 3*e*h) + a*c^2*g^2*h^2*(35*f*g^2 - 3*d*h^2))*x)*Sqrt[a + c*x^2])/
(8*h^5*(c*g^2 + a*h^2)^3*(g + h*x)^2) - ((4*c^2*f*g^5 - a^2*h^4*(2*f*g - 3*e*h) + a*c*g*h^2*(5*f*g^2 + 3*d*h^2
) + h*(4*a^2*f*h^4 + a*c*g*h^2*(14*f*g - 3*e*h) + c^2*(7*f*g^4 - 3*d*g^2*h^2))*x)*(a + c*x^2)^(3/2))/(12*h^3*(
c*g^2 + a*h^2)^2*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(5*h*(c*g^2 + a*h^2)*(g + h*x)^5)
+ (c^(3/2)*f*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/h^6 + (c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 + 3*a^3*h^6*(6
*f*g - e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*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)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.859968, antiderivative size = 507, 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 = {1651, 811, 844, 217, 206, 725} \[ -\frac{\left (a+c x^2\right )^{3/2} \left (h x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (3 d h^2+5 f g^2\right )+4 c^2 f g^5\right )}{12 h^3 (g+h x)^4 \left (a h^2+c g^2\right )^2}-\frac{c \sqrt{a+c x^2} \left (h x \left (a^2 c g h^4 (34 f g-3 e h)+8 a^3 f h^6+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )-a^3 h^6 (2 f g-3 e h)+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{8 h^5 (g+h x)^2 \left (a h^2+c g^2\right )^3}+\frac{c^2 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+3 a^3 h^6 (6 f g-e h)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{8 h^6 \left (a h^2+c g^2\right )^{7/2}}+\frac{c^{3/2} f \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]

[Out]

-(c*(8*c^3*f*g^7 + 20*a*c^2*f*g^5*h^2 - a^3*h^6*(2*f*g - 3*e*h) + a^2*c*g*h^4*(13*f*g^2 + 3*d*h^2) + h*(12*c^3
*f*g^6 + 8*a^3*f*h^6 + a^2*c*g*h^4*(34*f*g - 3*e*h) + a*c^2*g^2*h^2*(35*f*g^2 - 3*d*h^2))*x)*Sqrt[a + c*x^2])/
(8*h^5*(c*g^2 + a*h^2)^3*(g + h*x)^2) - ((4*c^2*f*g^5 - a^2*h^4*(2*f*g - 3*e*h) + a*c*g*h^2*(5*f*g^2 + 3*d*h^2
) + h*(4*a^2*f*h^4 + a*c*g*h^2*(14*f*g - 3*e*h) + c^2*(7*f*g^4 - 3*d*g^2*h^2))*x)*(a + c*x^2)^(3/2))/(12*h^3*(
c*g^2 + a*h^2)^2*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(5*h*(c*g^2 + a*h^2)*(g + h*x)^5)
+ (c^(3/2)*f*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/h^6 + (c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 + 3*a^3*h^6*(6
*f*g - e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*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)^(7/2))

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 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)^6} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}-\frac{\int \frac{\left (-5 (c d g-a f g+a e h)-5 f \left (\frac{c g^2}{h}+a h\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{5 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (4 c^2 f g^5-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (5 f g^2+3 d h^2\right )+h \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\int \frac{\left (-30 a c \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right )+\frac{40 c f \left (c g^2+a h^2\right )^2 x}{h}\right ) \sqrt{a+c x^2}}{(g+h x)^3} \, dx}{40 h^2 \left (c g^2+a h^2\right )^2}\\ &=-\frac{c \left (8 c^3 f g^7+20 a c^2 f g^5 h^2-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (13 f g^2+3 d h^2\right )+h \left (12 c^3 f g^6+8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac{\left (4 c^2 f g^5-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (5 f g^2+3 d h^2\right )+h \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}-\frac{\int \frac{20 a c^2 \left (4 c^2 f g^5+a^2 h^4 (10 f g-3 e h)+a c g h^2 \left (11 f g^2-3 d h^2\right )\right )-\frac{160 c^2 f \left (c g^2+a h^2\right )^3 x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{160 h^4 \left (c g^2+a h^2\right )^3}\\ &=-\frac{c \left (8 c^3 f g^7+20 a c^2 f g^5 h^2-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (13 f g^2+3 d h^2\right )+h \left (12 c^3 f g^6+8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac{\left (4 c^2 f g^5-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (5 f g^2+3 d h^2\right )+h \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (c^2 f\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{h^6}-\frac{\left (c^2 \left (8 c^3 f g^7+28 a c^2 f g^5 h^2+3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\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 )^3}\\ &=-\frac{c \left (8 c^3 f g^7+20 a c^2 f g^5 h^2-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (13 f g^2+3 d h^2\right )+h \left (12 c^3 f g^6+8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac{\left (4 c^2 f g^5-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (5 f g^2+3 d h^2\right )+h \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (c^2 f\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^2 \left (8 c^3 f g^7+28 a c^2 f g^5 h^2+3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\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 )^3}\\ &=-\frac{c \left (8 c^3 f g^7+20 a c^2 f g^5 h^2-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (13 f g^2+3 d h^2\right )+h \left (12 c^3 f g^6+8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac{\left (4 c^2 f g^5-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (5 f g^2+3 d h^2\right )+h \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{c^{3/2} f \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^6}+\frac{c^2 \left (8 c^3 f g^7+28 a c^2 f g^5 h^2+3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\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 )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 2.60789, size = 639, normalized size = 1.26 \[ \frac{-\frac{h \sqrt{a+c x^2} \left (2 (g+h x)^2 \left (a h^2+c g^2\right )^2 \left (20 a^2 f h^4+a c h^2 \left (3 h (8 d h-23 e g)+154 f g^2\right )+c^2 \left (9 g^2 h (3 d h-8 e g)+137 f g^4\right )\right )-c (g+h x)^3 \left (a h^2+c g^2\right ) \left (5 a^2 h^4 (58 f g-15 e h)+a c g h^2 \left (3 h (7 d h-62 e g)+631 f g^2\right )+c^2 \left (6 g^3 h (d h-16 e g)+326 f g^5\right )\right )+c (g+h x)^4 \left (3 a^2 c h^4 \left (h (8 d h-33 e g)+238 f g^2\right )+160 a^3 f h^6+3 a c^2 g^2 h^2 \left (261 f g^2-h (9 d h+26 e g)\right )+c^3 \left (274 f g^6-6 g^4 h (d h+4 e g)\right )\right )-6 (g+h x) \left (a h^2+c g^2\right )^3 \left (-5 a h^2 (e h-2 f g)+c g h (11 d h-16 e g)+21 c f g^3\right )+24 \left (a h^2+c g^2\right )^4 \left (h (d h-e g)+f g^2\right )\right )}{(g+h x)^5 \left (a h^2+c g^2\right )^3}+\frac{15 c^2 \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (a^2 c g h^4 \left (35 f g^2-3 d h^2\right )-3 a^3 h^6 (e h-6 f g)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{\left (a h^2+c g^2\right )^{7/2}}-\frac{15 c^2 \log (g+h x) \left (a^2 c g h^4 \left (35 f g^2-3 d h^2\right )-3 a^3 h^6 (e h-6 f g)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{\left (a h^2+c g^2\right )^{7/2}}+120 c^{3/2} f \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{120 h^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]

[Out]

(-((h*Sqrt[a + c*x^2]*(24*(c*g^2 + a*h^2)^4*(f*g^2 + h*(-(e*g) + d*h)) - 6*(c*g^2 + a*h^2)^3*(21*c*f*g^3 + c*g
*h*(-16*e*g + 11*d*h) - 5*a*h^2*(-2*f*g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^2*(20*a^2*f*h^4 + c^2*(137*f*g^4
+ 9*g^2*h*(-8*e*g + 3*d*h)) + a*c*h^2*(154*f*g^2 + 3*h*(-23*e*g + 8*d*h)))*(g + h*x)^2 - c*(c*g^2 + a*h^2)*(5
*a^2*h^4*(58*f*g - 15*e*h) + c^2*(326*f*g^5 + 6*g^3*h*(-16*e*g + d*h)) + a*c*g*h^2*(631*f*g^2 + 3*h*(-62*e*g +
7*d*h)))*(g + h*x)^3 + c*(160*a^3*f*h^6 + c^3*(274*f*g^6 - 6*g^4*h*(4*e*g + d*h)) + 3*a^2*c*h^4*(238*f*g^2 +
h*(-33*e*g + 8*d*h)) + 3*a*c^2*g^2*h^2*(261*f*g^2 - h*(26*e*g + 9*d*h)))*(g + h*x)^4))/((c*g^2 + a*h^2)^3*(g +
h*x)^5)) - (15*c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 - 3*a^3*h^6*(-6*f*g + e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d
*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(7/2) + 120*c^(3/2)*f*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^2*(8*c^3
*f*g^7 + 28*a*c^2*f*g^5*h^2 - 3*a^3*h^6*(-6*f*g + e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*Log[a*h - c*g*x + S
qrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/(120*h^6)

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Maple [B]  time = 0.26, size = 14169, normalized size = 28. \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)^6,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)^6,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)^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [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)**6,x)

[Out]

Timed out

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Giac [B]  time = 2.13869, size = 5951, normalized size = 11.74 \begin{align*} \text{result too large to display} \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)^6,x, algorithm="giac")

[Out]

-1/4*(8*c^5*f*g^7 + 28*a*c^4*f*g^5*h^2 + 35*a^2*c^3*f*g^3*h^4 - 3*a^2*c^3*d*g*h^6 + 18*a^3*c^2*f*g*h^6 - 3*a^3
*c^2*h^7*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^3*g^6*h^6 + 3*a*c^
2*g^4*h^8 + 3*a^2*c*g^2*h^10 + a^3*h^12)*sqrt(-c*g^2 - a*h^2)) - c^(3/2)*f*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a
)))/h^6 - 1/60*(600*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^5*f*g^7*h^4 + 1740*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^4
*f*g^5*h^6 + 1635*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*f*g^3*h^8 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c
^3*d*g*h^10 + 450*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^2*f*g*h^10 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^5*g
^6*h^5*e - 360*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^4*g^4*h^7*e - 360*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*g
^2*h^9*e - 75*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^2*h^11*e + 3600*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*f
*g^8*h^3 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*d*g^6*h^5 + 10020*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^
(9/2)*f*g^6*h^5 - 360*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*d*g^4*h^7 + 8595*(sqrt(c)*x - sqrt(c*x^2 + a))
^8*a^2*c^(7/2)*f*g^4*h^7 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d*g^2*h^9 + 1530*(sqrt(c)*x - sqrt(c
*x^2 + a))^8*a^3*c^(5/2)*f*g^2*h^9 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*d*h^11 - 240*(sqrt(c)*x -
sqrt(c*x^2 + a))^8*a^4*c^(3/2)*f*h^11 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*g^7*h^4*e - 1440*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*g^5*h^6*e - 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*g^3*h^8*e - 75
*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*g*h^10*e + 8800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*f*g^9*h^2 - 2
40*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*d*g^7*h^4 + 21240*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*f*g^7*h^4 - 720
*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*d*g^5*h^6 + 11670*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*f*g^5*h^6 + 6
90*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*d*g^3*h^8 - 4970*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*f*g^3*h^8
- 450*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*g*h^10 - 2580*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^2*f*g*h^10
- 960*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*g^8*h^3*e - 2640*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*g^6*h^5*e -
2160*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*g^4*h^7*e + 1170*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*g^2*h^9*
e + 30*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^2*h^11*e + 10000*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)*f*g^10*
h - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)*d*g^8*h^3 + 14040*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*
f*g^8*h^3 - 720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*d*g^6*h^5 - 14430*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a
^2*c^(9/2)*f*g^6*h^5 + 1590*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*d*g^4*h^7 - 28790*(sqrt(c)*x - sqrt(c*
x^2 + a))^6*a^3*c^(7/2)*f*g^4*h^7 - 1710*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*d*g^2*h^9 - 5820*(sqrt(c)
*x - sqrt(c*x^2 + a))^6*a^4*c^(5/2)*f*g^2*h^9 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(3/2)*f*h^11 - 960*(
sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)*g^9*h^2*e - 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*g^7*h^4*e
+ 720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*g^5*h^6*e + 4950*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)
*g^3*h^8*e - 270*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(5/2)*g*h^10*e + 4384*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c
^7*f*g^11 - 96*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*d*g^9*h^2 - 9392*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^6*f*g^
9*h^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^6*d*g^7*h^4 - 42996*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*f*g
^7*h^4 + 2364*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*d*g^5*h^6 - 31070*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^
4*f*g^5*h^6 - 2730*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*d*g^3*h^8 + 8620*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^
4*c^3*f*g^3*h^8 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*d*g*h^10 + 4800*(sqrt(c)*x - sqrt(c*x^2 + a))^5*
a^5*c^2*f*g*h^10 - 384*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*g^10*h*e + 672*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^
6*g^8*h^3*e + 3936*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*g^6*h^5*e + 5580*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^
3*c^4*g^4*h^7*e - 2970*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*g^2*h^9*e - 11920*(sqrt(c)*x - sqrt(c*x^2 + a))
^4*a*c^(13/2)*f*g^10*h + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d*g^8*h^3 - 15720*(sqrt(c)*x - sqrt(c*
x^2 + a))^4*a^2*c^(11/2)*f*g^8*h^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*d*g^6*h^5 + 19670*(sqrt(
c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*f*g^6*h^5 - 3510*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*d*g^4*h^7 +
36260*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*f*g^4*h^7 + 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2
)*d*g^2*h^9 + 6240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(5/2)*f*g^2*h^9 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4
*a^5*c^(5/2)*d*h^11 - 880*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(3/2)*f*h^11 + 960*(sqrt(c)*x - sqrt(c*x^2 + a
))^4*a*c^(13/2)*g^9*h^2*e + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*g^7*h^4*e - 480*(sqrt(c)*x - sqr
t(c*x^2 + a))^4*a^3*c^(9/2)*g^5*h^6*e - 6150*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*g^3*h^8*e + 720*(sqrt
(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(5/2)*g*h^10*e + 13120*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*f*g^9*h^2 - 24
0*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*d*g^7*h^4 + 30440*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*f*g^7*h^4
- 1200*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*d*g^5*h^6 + 14130*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*f*g^5
*h^6 + 2310*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*d*g^3*h^8 - 10790*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3*
f*g^3*h^8 - 510*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3*d*g*h^10 - 3820*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^
2*f*g*h^10 - 960*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*g^8*h^3*e - 2640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*
c^5*g^6*h^5*e - 2640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*g^4*h^7*e + 2790*(sqrt(c)*x - sqrt(c*x^2 + a))^3*
a^5*c^3*g^2*h^9*e - 30*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^2*h^11*e - 7360*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a
^3*c^(11/2)*f*g^8*h^3 + 120*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*d*g^6*h^5 - 19930*(sqrt(c)*x - sqrt(c
*x^2 + a))^2*a^4*c^(9/2)*f*g^6*h^5 + 690*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(9/2)*d*g^4*h^7 - 16050*(sqrt(c
)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*f*g^4*h^7 - 1050*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*d*g^2*h^9 -
1300*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(5/2)*f*g^2*h^9 + 560*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(3/2)*f
*h^11 + 480*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*g^7*h^4*e + 1560*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*
c^(9/2)*g^5*h^6*e + 2130*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*g^3*h^8*e - 570*(sqrt(c)*x - sqrt(c*x^2 +
a))^2*a^6*c^(5/2)*g*h^10*e + 2140*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c^5*f*g^7*h^4 - 60*(sqrt(c)*x - sqrt(c*x^
2 + a))*a^4*c^5*d*g^5*h^6 + 6090*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^4*f*g^5*h^6 - 270*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^5*c^4*d*g^3*h^8 + 5505*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^3*f*g^3*h^8 + 195*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^6*c^3*d*g*h^10 + 1150*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^2*f*g*h^10 - 120*(sqrt(c)*x - sqrt(c*x^2 + a
))*a^4*c^5*g^6*h^5*e - 420*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^4*g^4*h^7*e - 630*(sqrt(c)*x - sqrt(c*x^2 + a))
*a^6*c^3*g^2*h^9*e + 75*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^2*h^11*e - 274*a^5*c^(9/2)*f*g^6*h^5 + 6*a^5*c^(9/
2)*d*g^4*h^7 - 783*a^6*c^(7/2)*f*g^4*h^7 + 27*a^6*c^(7/2)*d*g^2*h^9 - 714*a^7*c^(5/2)*f*g^2*h^9 - 24*a^7*c^(5/
2)*d*h^11 - 160*a^8*c^(3/2)*f*h^11 + 24*a^5*c^(9/2)*g^5*h^6*e + 78*a^6*c^(7/2)*g^3*h^8*e + 99*a^7*c^(5/2)*g*h^
10*e)/((c^3*g^6*h^6 + 3*a*c^2*g^4*h^8 + 3*a^2*c*g^2*h^10 + a^3*h^12)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(s
qrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g - a*h)^5)