3.1.99 \(\int \frac {(a+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^8} \, dx\) [99]

Optimal. Leaf size=532 \[ -\frac {a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac {c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (7 a h^2 (2 f g-e h)+c g \left (5 f g^2+h (2 e g-9 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac {\left (42 a^2 f h^4-c^2 g^2 \left (5 f g^2+h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}-\frac {a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{16 \left (c g^2+a h^2\right )^{11/2}} \]

[Out]

-1/24*c*(6*c^2*d*g^3+a^2*h^2*(-e*h+8*f*g)-a*c*g*(f*g^2-h*(-3*d*h+8*e*g)))*(-c*g*x+a*h)*(c*x^2+a)^(3/2)/(a*h^2+
c*g^2)^4/(h*x+g)^4-1/7*(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(5/2)/h/(a*h^2+c*g^2)/(h*x+g)^7+1/42*(7*a*h^2*(-e*h+2*f*g
)+c*g*(5*f*g^2+h*(-9*d*h+2*e*g)))*(c*x^2+a)^(5/2)/h/(a*h^2+c*g^2)^2/(h*x+g)^6-1/210*(42*a^2*f*h^4-c^2*g^2*(5*f
*g^2+h*(-51*d*h+2*e*g))-a*c*h^2*(26*f*g^2-h*(-12*d*h+61*e*g)))*(c*x^2+a)^(5/2)/h/(a*h^2+c*g^2)^3/(h*x+g)^5-1/1
6*a^2*c^3*(6*c^2*d*g^3+a^2*h^2*(-e*h+8*f*g)-a*c*g*(f*g^2-h*(-3*d*h+8*e*g)))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)
^(1/2)/(c*x^2+a)^(1/2))/(a*h^2+c*g^2)^(11/2)-1/16*a*c^2*(6*c^2*d*g^3+a^2*h^2*(-e*h+8*f*g)-a*c*g*(f*g^2-h*(-3*d
*h+8*e*g)))*(-c*g*x+a*h)*(c*x^2+a)^(1/2)/(a*h^2+c*g^2)^5/(h*x+g)^2

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Rubi [A]
time = 0.55, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1665, 849, 821, 735, 739, 212} \begin {gather*} -\frac {\left (a+c x^2\right )^{5/2} \left (42 a^2 f h^4-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )-c^2 \left (g^2 h (2 e g-51 d h)+5 f g^4\right )\right )}{210 h (g+h x)^5 \left (a h^2+c g^2\right )^3}-\frac {a c^2 \sqrt {a+c x^2} (a h-c g x) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{16 (g+h x)^2 \left (a h^2+c g^2\right )^5}-\frac {c \left (a+c x^2\right )^{3/2} (a h-c g x) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{24 (g+h x)^4 \left (a h^2+c g^2\right )^4}-\frac {a^2 c^3 \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 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{16 \left (a h^2+c g^2\right )^{11/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{7 h (g+h x)^7 \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{5/2} \left (7 a h^2 (2 f g-e h)+c g h (2 e g-9 d h)+5 c f g^3\right )}{42 h (g+h x)^6 \left (a h^2+c g^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]

[Out]

-1/16*(a*c^2*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*d*h)))*(a*h - c*g*x)*Sqrt[a +
c*x^2])/((c*g^2 + a*h^2)^5*(g + h*x)^2) - (c*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g -
3*d*h)))*(a*h - c*g*x)*(a + c*x^2)^(3/2))/(24*(c*g^2 + a*h^2)^4*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c
*x^2)^(5/2))/(7*h*(c*g^2 + a*h^2)*(g + h*x)^7) + ((5*c*f*g^3 + c*g*h*(2*e*g - 9*d*h) + 7*a*h^2*(2*f*g - e*h))*
(a + c*x^2)^(5/2))/(42*h*(c*g^2 + a*h^2)^2*(g + h*x)^6) - ((42*a^2*f*h^4 - c^2*(5*f*g^4 + g^2*h*(2*e*g - 51*d*
h)) - a*c*h^2*(26*f*g^2 - h*(61*e*g - 12*d*h)))*(a + c*x^2)^(5/2))/(210*h*(c*g^2 + a*h^2)^3*(g + h*x)^5) - (a^
2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*
g^2 + a*h^2]*Sqrt[a + c*x^2])])/(16*(c*g^2 + a*h^2)^(11/2))

Rule 212

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

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 0]

Rule 739

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]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(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
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

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

Rule 1665

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]

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)^8} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}-\frac {\int \frac {\left (-7 (c d g-a f g+a e h)-\left (7 a f h+c \left (2 e g+\frac {5 f g^2}{h}-2 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^7} \, dx}{7 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}+\frac {\int \frac {\left (6 \left (7 c^2 d g^2+7 a^2 f h^2-a c \left (2 f g^2-h (9 e g-2 d h)\right )\right )+\frac {c \left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) x}{h}\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^6} \, dx}{42 \left (c g^2+a h^2\right )^2}\\ &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac {\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}+\frac {\left (c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right )\right ) \int \frac {\left (a+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{6 \left (c g^2+a h^2\right )^3}\\ &=-\frac {c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac {\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}+\frac {\left (a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right )\right ) \int \frac {\sqrt {a+c x^2}}{(g+h x)^3} \, dx}{8 \left (c g^2+a h^2\right )^4}\\ &=-\frac {a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac {c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac {\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}+\frac {\left (a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{16 \left (c g^2+a h^2\right )^5}\\ &=-\frac {a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac {c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac {\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}-\frac {\left (a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right )\right ) \text {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 )}{16 \left (c g^2+a h^2\right )^5}\\ &=-\frac {a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac {c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac {\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac {\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}-\frac {a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )}{16 \left (c g^2+a h^2\right )^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 11.61, size = 863, normalized size = 1.62 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (240 \left (c g^2+a h^2\right )^6 \left (f g^2+h (-e g+d h)\right )-40 \left (c g^2+a h^2\right )^5 \left (29 c f g^3+c g h (-22 e g+15 d h)-7 a h^2 (-2 f g+e h)\right ) (g+h x)+8 \left (c g^2+a h^2\right )^4 \left (42 a^2 f h^4+a c h^2 \left (314 f g^2+h (-139 e g+48 d h)\right )+c^2 \left (275 f g^4+g^2 h (-142 e g+51 d h)\right )\right ) (g+h x)^2-2 c \left (c g^2+a h^2\right )^3 \left (7 a^2 h^4 (136 f g-35 e h)+2 c^2 \left (500 f g^5+g^3 h (-136 e g+3 d h)\right )+a c g h^2 \left (1979 f g^2+h (-544 e g+33 d h)\right )\right ) (g+h x)^3+2 c \left (c g^2+a h^2\right )^2 \left (336 a^3 f h^6+c^3 \left (400 f g^6-2 g^4 h (4 e g+3 d h)\right )+3 a^2 c h^4 \left (400 f g^2+h (-29 e g+8 d h)\right )+a c^2 g^2 h^2 \left (1201 f g^2-h (32 e g+45 d h)\right )\right ) (g+h x)^4-c^2 \left (c g^2+a h^2\right ) \left (21 a^3 h^6 (24 f g-5 e h)+2 a c^2 g^3 h^2 \left (89 f g^2+44 e g h+54 d h^2\right )+3 a^2 c g h^4 \left (109 f g^2+h (94 e g-73 d h)\right )+4 c^3 \left (10 f g^7+g^5 h (4 e g+3 d h)\right )\right ) (g+h x)^5-c^2 \left (-336 a^4 f h^8+2 a c^3 g^4 h^2 \left (109 f g^2+52 e g h+60 d h^2\right )+a^2 c^2 g^2 h^4 \left (505 f g^2+h (370 e g-741 d h)\right )+4 c^4 \left (10 f g^8+g^6 h (4 e g+3 d h)\right )+3 a^3 c h^6 \left (312 f g^2+h (-221 e g+32 d h)\right )\right ) (g+h x)^6\right )}{1680 \left (c g^2 h+a h^3\right )^5 (g+h x)^7}+\frac {a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2+h (-8 e g+3 d h)\right )\right ) \log (g+h x)}{16 \left (c g^2+a h^2\right )^{11/2}}-\frac {a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2+h (-8 e g+3 d h)\right )\right ) \log \left (a h-c g x+\sqrt {c g^2+a h^2} \sqrt {a+c x^2}\right )}{16 \left (c g^2+a h^2\right )^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]

[Out]

-1/1680*(Sqrt[a + c*x^2]*(240*(c*g^2 + a*h^2)^6*(f*g^2 + h*(-(e*g) + d*h)) - 40*(c*g^2 + a*h^2)^5*(29*c*f*g^3
+ c*g*h*(-22*e*g + 15*d*h) - 7*a*h^2*(-2*f*g + e*h))*(g + h*x) + 8*(c*g^2 + a*h^2)^4*(42*a^2*f*h^4 + a*c*h^2*(
314*f*g^2 + h*(-139*e*g + 48*d*h)) + c^2*(275*f*g^4 + g^2*h*(-142*e*g + 51*d*h)))*(g + h*x)^2 - 2*c*(c*g^2 + a
*h^2)^3*(7*a^2*h^4*(136*f*g - 35*e*h) + 2*c^2*(500*f*g^5 + g^3*h*(-136*e*g + 3*d*h)) + a*c*g*h^2*(1979*f*g^2 +
h*(-544*e*g + 33*d*h)))*(g + h*x)^3 + 2*c*(c*g^2 + a*h^2)^2*(336*a^3*f*h^6 + c^3*(400*f*g^6 - 2*g^4*h*(4*e*g
+ 3*d*h)) + 3*a^2*c*h^4*(400*f*g^2 + h*(-29*e*g + 8*d*h)) + a*c^2*g^2*h^2*(1201*f*g^2 - h*(32*e*g + 45*d*h)))*
(g + h*x)^4 - c^2*(c*g^2 + a*h^2)*(21*a^3*h^6*(24*f*g - 5*e*h) + 2*a*c^2*g^3*h^2*(89*f*g^2 + 44*e*g*h + 54*d*h
^2) + 3*a^2*c*g*h^4*(109*f*g^2 + h*(94*e*g - 73*d*h)) + 4*c^3*(10*f*g^7 + g^5*h*(4*e*g + 3*d*h)))*(g + h*x)^5
- c^2*(-336*a^4*f*h^8 + 2*a*c^3*g^4*h^2*(109*f*g^2 + 52*e*g*h + 60*d*h^2) + a^2*c^2*g^2*h^4*(505*f*g^2 + h*(37
0*e*g - 741*d*h)) + 4*c^4*(10*f*g^8 + g^6*h*(4*e*g + 3*d*h)) + 3*a^3*c*h^6*(312*f*g^2 + h*(-221*e*g + 32*d*h))
)*(g + h*x)^6))/((c*g^2*h + a*h^3)^5*(g + h*x)^7) + (a^2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g
^2 + h*(-8*e*g + 3*d*h)))*Log[g + h*x])/(16*(c*g^2 + a*h^2)^(11/2)) - (a^2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g -
e*h) - a*c*g*(f*g^2 + h*(-8*e*g + 3*d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(16*(c*g^2
+ a*h^2)^(11/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(24804\) vs. \(2(504)=1008\).
time = 0.10, size = 24805, normalized size = 46.63

method	result	size
default	\(\text {Expression too large to display}\)	\(24805\)

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)^8,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 16286 vs. \(2 (514) = 1028\).
time = 0.92, size = 16286, normalized size = 30.61 \begin {gather*} \text {Too large to display} \end {gather*}

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)^8,x, algorithm="maxima")

[Out]

9/16*sqrt(c*x^2 + a)*c^7*f*g^9/(c^6*g^12*h^5 + 6*a*c^5*g^10*h^7 + 15*a^2*c^4*g^8*h^9 + 20*a^3*c^3*g^6*h^11 + 1
5*a^4*c^2*g^4*h^13 + 6*a^5*c*g^2*h^15 + a^6*h^17) - 9/16*sqrt(c*x^2 + a)*c^7*f*g^8*x/(c^6*g^12*h^4 + 6*a*c^5*g
^10*h^6 + 15*a^2*c^4*g^8*h^8 + 20*a^3*c^3*g^6*h^10 + 15*a^4*c^2*g^4*h^12 + 6*a^5*c*g^2*h^14 + a^6*h^16) + 3/16
*(c*x^2 + a)^(3/2)*c^6*f*g^8/(c^6*g^12*h^4*x + 6*a*c^5*g^10*h^6*x + 15*a^2*c^4*g^8*h^8*x + 20*a^3*c^3*g^6*h^10
*x + 15*a^4*c^2*g^4*h^12*x + 6*a^5*c*g^2*h^14*x + a^6*h^16*x + c^6*g^13*h^3 + 6*a*c^5*g^11*h^5 + 15*a^2*c^4*g^
9*h^7 + 20*a^3*c^3*g^7*h^9 + 15*a^4*c^2*g^5*h^11 + 6*a^5*c*g^3*h^13 + a^6*g*h^15) - 9/16*sqrt(c*x^2 + a)*c^7*g
^8*e/(c^6*g^12*h^4 + 6*a*c^5*g^10*h^6 + 15*a^2*c^4*g^8*h^8 + 20*a^3*c^3*g^6*h^10 + 15*a^4*c^2*g^4*h^12 + 6*a^5
*c*g^2*h^14 + a^6*h^16) + 9/16*sqrt(c*x^2 + a)*c^7*g^7*x*e/(c^6*g^12*h^3 + 6*a*c^5*g^10*h^5 + 15*a^2*c^4*g^8*h
^7 + 20*a^3*c^3*g^6*h^9 + 15*a^4*c^2*g^4*h^11 + 6*a^5*c*g^2*h^13 + a^6*h^15) + 9/16*sqrt(c*x^2 + a)*c^7*d*g^7/
(c^6*g^12*h^3 + 6*a*c^5*g^10*h^5 + 15*a^2*c^4*g^8*h^7 + 20*a^3*c^3*g^6*h^9 + 15*a^4*c^2*g^4*h^11 + 6*a^5*c*g^2
*h^13 + a^6*h^15) - 3/16*(c*x^2 + a)^(5/2)*c^5*f*g^7/(c^6*g^12*h^3*x^2 + 6*a*c^5*g^10*h^5*x^2 + 15*a^2*c^4*g^8
*h^7*x^2 + 20*a^3*c^3*g^6*h^9*x^2 + 15*a^4*c^2*g^4*h^11*x^2 + 6*a^5*c*g^2*h^13*x^2 + a^6*h^15*x^2 + 2*c^6*g^13
*h^2*x + 12*a*c^5*g^11*h^4*x + 30*a^2*c^4*g^9*h^6*x + 40*a^3*c^3*g^7*h^8*x + 30*a^4*c^2*g^5*h^10*x + 12*a^5*c*
g^3*h^12*x + 2*a^6*g*h^14*x + c^6*g^14*h + 6*a*c^5*g^12*h^3 + 15*a^2*c^4*g^10*h^5 + 20*a^3*c^3*g^8*h^7 + 15*a^
4*c^2*g^6*h^9 + 6*a^5*c*g^4*h^11 + a^6*g^2*h^13) + 3/16*(c*x^2 + a)^(3/2)*c^6*f*g^7/(c^6*g^12*h^3 + 6*a*c^5*g^
10*h^5 + 15*a^2*c^4*g^8*h^7 + 20*a^3*c^3*g^6*h^9 + 15*a^4*c^2*g^4*h^11 + 6*a^5*c*g^2*h^13 + a^6*h^15) - 9/16*s
qrt(c*x^2 + a)*c^7*d*g^6*x/(c^6*g^12*h^2 + 6*a*c^5*g^10*h^4 + 15*a^2*c^4*g^8*h^6 + 20*a^3*c^3*g^6*h^8 + 15*a^4
*c^2*g^4*h^10 + 6*a^5*c*g^2*h^12 + a^6*h^14) - 3/16*(c*x^2 + a)^(3/2)*c^6*g^7*e/(c^6*g^12*h^3*x + 6*a*c^5*g^10
*h^5*x + 15*a^2*c^4*g^8*h^7*x + 20*a^3*c^3*g^6*h^9*x + 15*a^4*c^2*g^4*h^11*x + 6*a^5*c*g^2*h^13*x + a^6*h^15*x
+ c^6*g^13*h^2 + 6*a*c^5*g^11*h^4 + 15*a^2*c^4*g^9*h^6 + 20*a^3*c^3*g^7*h^8 + 15*a^4*c^2*g^5*h^10 + 6*a^5*c*g
^3*h^12 + a^6*g*h^14) + 3/16*(c*x^2 + a)^(3/2)*c^6*d*g^6/(c^6*g^12*h^2*x + 6*a*c^5*g^10*h^4*x + 15*a^2*c^4*g^8
*h^6*x + 20*a^3*c^3*g^6*h^8*x + 15*a^4*c^2*g^4*h^10*x + 6*a^5*c*g^2*h^12*x + a^6*h^14*x + c^6*g^13*h + 6*a*c^5
*g^11*h^3 + 15*a^2*c^4*g^9*h^5 + 20*a^3*c^3*g^7*h^7 + 15*a^4*c^2*g^5*h^9 + 6*a^5*c*g^3*h^11 + a^6*g*h^13) - 35
/16*sqrt(c*x^2 + a)*c^6*f*g^7/(c^5*g^10*h^5 + 5*a*c^4*g^8*h^7 + 10*a^2*c^3*g^6*h^9 + 10*a^3*c^2*g^4*h^11 + 5*a
^4*c*g^2*h^13 + a^5*h^15) + 13/8*sqrt(c*x^2 + a)*c^6*f*g^6*x/(c^5*g^10*h^4 + 5*a*c^4*g^8*h^6 + 10*a^2*c^3*g^6*
h^8 + 10*a^3*c^2*g^4*h^10 + 5*a^4*c*g^2*h^12 + a^5*h^14) + 3/16*(c*x^2 + a)^(5/2)*c^5*g^6*e/(c^6*g^12*h^2*x^2
+ 6*a*c^5*g^10*h^4*x^2 + 15*a^2*c^4*g^8*h^6*x^2 + 20*a^3*c^3*g^6*h^8*x^2 + 15*a^4*c^2*g^4*h^10*x^2 + 6*a^5*c*g
^2*h^12*x^2 + a^6*h^14*x^2 + 2*c^6*g^13*h*x + 12*a*c^5*g^11*h^3*x + 30*a^2*c^4*g^9*h^5*x + 40*a^3*c^3*g^7*h^7*
x + 30*a^4*c^2*g^5*h^9*x + 12*a^5*c*g^3*h^11*x + 2*a^6*g*h^13*x + c^6*g^14 + 6*a*c^5*g^12*h^2 + 15*a^2*c^4*g^1
0*h^4 + 20*a^3*c^3*g^8*h^6 + 15*a^4*c^2*g^6*h^8 + 6*a^5*c*g^4*h^10 + a^6*g^2*h^12) - 3/16*(c*x^2 + a)^(3/2)*c^
6*g^6*e/(c^6*g^12*h^2 + 6*a*c^5*g^10*h^4 + 15*a^2*c^4*g^8*h^6 + 20*a^3*c^3*g^6*h^8 + 15*a^4*c^2*g^4*h^10 + 6*a
^5*c*g^2*h^12 + a^6*h^14) - 3/16*(c*x^2 + a)^(5/2)*c^5*d*g^5/(c^6*g^12*h*x^2 + 6*a*c^5*g^10*h^3*x^2 + 15*a^2*c
^4*g^8*h^5*x^2 + 20*a^3*c^3*g^6*h^7*x^2 + 15*a^4*c^2*g^4*h^9*x^2 + 6*a^5*c*g^2*h^11*x^2 + a^6*h^13*x^2 + 2*c^6
*g^13*x + 12*a*c^5*g^11*h^2*x + 30*a^2*c^4*g^9*h^4*x + 40*a^3*c^3*g^7*h^6*x + 30*a^4*c^2*g^5*h^8*x + 12*a^5*c*
g^3*h^10*x + 2*a^6*g*h^12*x + c^6*g^14/h + 6*a*c^5*g^12*h + 15*a^2*c^4*g^10*h^3 + 20*a^3*c^3*g^8*h^5 + 15*a^4*
c^2*g^6*h^7 + 6*a^5*c*g^4*h^9 + a^6*g^2*h^11) + 3/16*(c*x^2 + a)^(3/2)*c^6*d*g^5/(c^6*g^12*h + 6*a*c^5*g^10*h^
3 + 15*a^2*c^4*g^8*h^5 + 20*a^3*c^3*g^6*h^7 + 15*a^4*c^2*g^4*h^9 + 6*a^5*c*g^2*h^11 + a^6*h^13) - 3/8*(c*x^2 +
a)^(5/2)*c^4*f*g^6/(c^5*g^10*h^4*x^3 + 5*a*c^4*g^8*h^6*x^3 + 10*a^2*c^3*g^6*h^8*x^3 + 10*a^3*c^2*g^4*h^10*x^3
+ 5*a^4*c*g^2*h^12*x^3 + a^5*h^14*x^3 + 3*c^5*g^11*h^3*x^2 + 15*a*c^4*g^9*h^5*x^2 + 30*a^2*c^3*g^7*h^7*x^2 +
30*a^3*c^2*g^5*h^9*x^2 + 15*a^4*c*g^3*h^11*x^2 + 3*a^5*g*h^13*x^2 + 3*c^5*g^12*h^2*x + 15*a*c^4*g^10*h^4*x + 3
0*a^2*c^3*g^8*h^6*x + 30*a^3*c^2*g^6*h^8*x + 15*a^4*c*g^4*h^10*x + 3*a^5*g^2*h^12*x + c^5*g^13*h + 5*a*c^4*g^1
1*h^3 + 10*a^2*c^3*g^9*h^5 + 10*a^3*c^2*g^7*h^7 + 5*a^4*c*g^5*h^9 + a^5*g^3*h^11) - 11/12*(c*x^2 + a)^(3/2)*c^
5*f*g^6/(c^5*g^10*h^4*x + 5*a*c^4*g^8*h^6*x + 10*a^2*c^3*g^6*h^8*x + 10*a^3*c^2*g^4*h^10*x + 5*a^4*c*g^2*h^12*
x + a^5*h^14*x + c^5*g^11*h^3 + 5*a*c^4*g^9*h^5 + 10*a^2*c^3*g^7*h^7 + 10*a^3*c^2*g^5*h^9 + 5*a^4*c*g^3*h^11 +
a^5*g*h^13) + 7/4*sqrt(c*x^2 + a)*c^6*g^6*e/(c^5*g^10*h^4 + 5*a*c^4*g^8*h^6 + 10*a^2*c^3*g^6*h^8 + 10*a^3*c^2
*g^4*h^10 + 5*a^4*c*g^2*h^12 + a^5*h^14) - 19/1...

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)^8,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7936 vs. \(2 (514) = 1028\).
time = 7.84, size = 7936, normalized size = 14.92 \begin {gather*} \text {Too large to display} \end {gather*}

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)^8,x, algorithm="giac")

[Out]

-1/8*(6*a^2*c^5*d*g^3 - a^3*c^4*f*g^3 - 3*a^3*c^4*d*g*h^2 + 8*a^4*c^3*f*g*h^2 + 8*a^3*c^4*g^2*h*e - a^4*c^3*h^
3*e)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^5*g^10 + 5*a*c^4*g^8*h^2 +
10*a^2*c^3*g^6*h^4 + 10*a^3*c^2*g^4*h^6 + 5*a^4*c*g^2*h^8 + a^5*h^10)*sqrt(-c*g^2 - a*h^2)) - 1/840*(630*(sqr
t(c)*x - sqrt(c*x^2 + a))^13*a^2*c^5*d*g^3*h^12 - 105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*f*g^3*h^12 - 31
5*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*d*g*h^14 + 840*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^4*c^3*f*g*h^14 +
840*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*g^2*h^13*e - 105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^4*c^3*h^15*e
- 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^12*c^(15/2)*f*g^10*h^5 - 8400*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a*c^(13/2)
*f*g^8*h^7 - 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^2*c^(11/2)*f*g^6*h^9 + 8190*(sqrt(c)*x - sqrt(c*x^2 + a)
)^12*a^2*c^(11/2)*d*g^4*h^11 - 18165*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)*f*g^4*h^11 - 4095*(sqrt(c)*x
- sqrt(c*x^2 + a))^12*a^3*c^(9/2)*d*g^2*h^13 + 2520*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*f*g^2*h^13 -
1680*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^5*c^(5/2)*f*h^15 + 10920*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)
*g^3*h^12*e - 1365*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*g*h^14*e - 5600*(sqrt(c)*x - sqrt(c*x^2 + a))^
11*c^8*f*g^11*h^4 - 28000*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*f*g^9*h^6 - 56000*(sqrt(c)*x - sqrt(c*x^2 + a
))^11*a^2*c^6*f*g^7*h^8 + 44940*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*d*g^5*h^10 - 63490*(sqrt(c)*x - sqrt(
c*x^2 + a))^11*a^3*c^5*f*g^5*h^10 - 26670*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*d*g^3*h^12 + 32620*(sqrt(c)
*x - sqrt(c*x^2 + a))^11*a^4*c^4*f*g^3*h^12 + 2100*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^4*c^4*d*g*h^14 - 11200*(
sqrt(c)*x - sqrt(c*x^2 + a))^11*a^5*c^3*f*g*h^14 - 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^11*c^8*g^10*h^5*e - 1120
0*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*g^8*h^7*e - 22400*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*g^6*h^9*e
+ 37520*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*g^4*h^11*e - 24290*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^4*c^4*g
^2*h^13*e - 1540*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^5*c^3*h^15*e - 11200*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(1
7/2)*f*g^12*h^3 - 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*d*g^10*h^5 - 52640*(sqrt(c)*x - sqrt(c*x^2 +
a))^10*a*c^(15/2)*f*g^10*h^5 - 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/2)*d*g^8*h^7 - 95200*(sqrt(c)*x
- sqrt(c*x^2 + a))^10*a^2*c^(13/2)*f*g^8*h^7 + 100380*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*d*g^6*h^9
- 100730*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(11/2)*f*g^6*h^9 - 146790*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3
*c^(11/2)*d*g^4*h^11 + 163940*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(9/2)*f*g^4*h^11 + 6300*(sqrt(c)*x - sqrt
(c*x^2 + a))^10*a^4*c^(9/2)*d*g^2*h^13 - 56000*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(7/2)*f*g^2*h^13 - 3360*
(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(7/2)*d*h^15 + 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^6*c^(5/2)*f*h^15
- 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*g^11*h^4*e - 22400*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/
2)*g^9*h^6*e - 44800*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*g^7*h^8*e + 133840*(sqrt(c)*x - sqrt(c*x^2
+ a))^10*a^3*c^(11/2)*g^5*h^10*e - 106330*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(9/2)*g^3*h^12*e + 3220*(sqrt
(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(7/2)*g*h^14*e - 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*f*g^13*h^2 - 4032
*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*d*g^11*h^4 - 50848*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^8*f*g^11*h^4 - 201
60*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^8*d*g^9*h^6 - 52640*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*f*g^9*h^6 +
191016*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*d*g^7*h^8 - 9436*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^6*f*g^7
*h^8 - 363216*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^6*d*g^5*h^10 + 439306*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*
c^5*f*g^5*h^10 + 95340*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^5*d*g^3*h^12 - 209965*(sqrt(c)*x - sqrt(c*x^2 + a
))^9*a^5*c^4*f*g^3*h^12 - 9975*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^5*c^4*d*g*h^14 + 32200*(sqrt(c)*x - sqrt(c*x^
2 + a))^9*a^6*c^3*f*g*h^14 - 5376*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*g^12*h^3*e - 25984*(sqrt(c)*x - sqrt(c*x
^2 + a))^9*a*c^8*g^10*h^5*e - 49280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*g^8*h^7*e + 263648*(sqrt(c)*x - sq
rt(c*x^2 + a))^9*a^3*c^6*g^6*h^9*e - 332780*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^5*g^4*h^11*e + 49490*(sqrt(c
)*x - sqrt(c*x^2 + a))^9*a^5*c^4*g^2*h^13*e - 1085*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^6*c^3*h^15*e - 8960*(sqrt
(c)*x - sqrt(c*x^2 + a))^8*c^(19/2)*f*g^14*h - 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(19/2)*d*g^12*h^3 - 1523
2*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(17/2)*f*g^12*h^3 - 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(17/2)*d*g
^10*h^5 + 53200*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(15/2)*f*g^10*h^5 + 181104*(sqrt(c)*x - sqrt(c*x^2 + a))
^8*a^2*c^(15/2)*d*g^8*h^7 + 143416*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(13/2)*f*g^8*h^7 - 651924*(sqrt(c)*x
- sqrt(c*x^2 + a))^8*a^3*c^(13/2)*d*g^6*h^9 + 5...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x)

[Out]

int(((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8, x)

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