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

[Out]

1/24*(4*a^2*h^4*(-2*e*h+f*g)-4*c^2*g^4*(-e*h+5*f*g)-a*c*g*h^2*(25*f*g^2-h*(-9*d*h+5*e*g))-3*h*(4*a^2*f*h^4+a*c
*h^2*(17*f*g^2-h*(-d*h+5*e*g))+2*c^2*g^2*(5*f*g^2-h*(d*h+e*g)))*x)*(c*x^2+a)^(3/2)/h^3/(a*h^2+c*g^2)^2/(h*x+g)
^3-1/4*(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)^4-c^(3/2)*(-e*h+5*f*g)*arctanh(x*c^(1/2)/(c
*x^2+a)^(1/2))/h^6-1/8*c*(12*a^3*f*h^6+8*c^3*g^5*(-e*h+5*f*g)+20*a*c^2*g^3*h^2*(-e*h+5*f*g)+3*a^2*c*h^4*(25*f*
g^2-h*(-d*h+5*e*g)))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/h^6/(a*h^2+c*g^2)^(5/2)+1/8*c*(
8*(-e*h+5*f*g)*(a*h^2+c*g^2)^2+h*(12*a^2*f*h^4+4*c^2*g^3*(-e*h+5*f*g)+a*c*h^2*(35*f*g^2-h*(-3*d*h+7*e*g)))*x)*
(c*x^2+a)^(1/2)/h^5/(a*h^2+c*g^2)^2/(h*x+g)

________________________________________________________________________________________

Rubi [A]  time = 1.09, antiderivative size = 511, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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]

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 = 2.15, size = 575, normalized size = 1.13 \[ -\frac {\frac {h \sqrt {a+c x^2} \left ((g+h x)^2 \left (a h^2+c g^2\right ) \left (12 a^2 f h^4+a c h^2 \left (h (15 d h-43 e g)+95 f g^2\right )+2 c^2 \left (g^2 h (9 d h-23 e g)+43 f g^4\right )\right )-c (g+h x)^3 \left (4 a^2 h^4 (31 f g-8 e h)+a c g h^2 \left (h (15 d h-91 e g)+287 f g^2\right )+2 c^2 \left (g^3 h (3 d h-25 e g)+77 f g^5\right )\right )+6 \left (a h^2+c g^2\right )^3 \left (h (d h-e g)+f g^2\right )-2 (g+h x) \left (a h^2+c g^2\right )^2 \left (-4 a h^2 (e h-2 f g)+c g h (9 d h-13 e g)+17 c f g^3\right )-24 c f (g+h x)^4 \left (a h^2+c g^2\right )^2\right )}{(g+h x)^4 \left (a h^2+c g^2\right )^2}+\frac {3 c \log \left (\sqrt {a+c x^2} \sqrt {a h^2+c g^2}+a h-c g x\right ) \left (12 a^3 f h^6+3 a^2 c h^4 \left (h (d h-5 e g)+25 f g^2\right )+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{\left (a h^2+c g^2\right )^{5/2}}-\frac {3 c \log (g+h x) \left (12 a^3 f h^6+3 a^2 c h^4 \left (h (d h-5 e g)+25 f g^2\right )+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{\left (a h^2+c g^2\right )^{5/2}}+24 c^{3/2} \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right ) (5 f g-e h)}{24 h^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*((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))/h^6

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

maple [B]  time = 0.02, size = 12481, normalized size = 24.42 \[ \text {output too large to display} \]

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

________________________________________________________________________________________

maxima [B]  time = 1.50, size = 4326, normalized size = 8.47 \[ \text {result too large to display} \]

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]

3/8*sqrt(c*x^2 + a)*c^4*f*g^6/(c^3*g^6*h^5 + 3*a*c^2*g^4*h^7 + 3*a^2*c*g^2*h^9 + a^3*h^11) - 3/8*sqrt(c*x^2 +
a)*c^4*f*g^5*x/(c^3*g^6*h^4 + 3*a*c^2*g^4*h^6 + 3*a^2*c*g^2*h^8 + a^3*h^10) - 3/8*sqrt(c*x^2 + a)*c^4*e*g^5/(c
^3*g^6*h^4 + 3*a*c^2*g^4*h^6 + 3*a^2*c*g^2*h^8 + a^3*h^10) + 1/8*(c*x^2 + a)^(3/2)*c^3*f*g^5/(c^3*g^6*h^4*x +
3*a*c^2*g^4*h^6*x + 3*a^2*c*g^2*h^8*x + a^3*h^10*x + c^3*g^7*h^3 + 3*a*c^2*g^5*h^5 + 3*a^2*c*g^3*h^7 + a^3*g*h
^9) + 3/8*sqrt(c*x^2 + a)*c^4*e*g^4*x/(c^3*g^6*h^3 + 3*a*c^2*g^4*h^5 + 3*a^2*c*g^2*h^7 + a^3*h^9) + 3/8*sqrt(c
*x^2 + a)*c^4*d*g^4/(c^3*g^6*h^3 + 3*a*c^2*g^4*h^5 + 3*a^2*c*g^2*h^7 + a^3*h^9) - 1/8*(c*x^2 + a)^(3/2)*c^3*e*
g^4/(c^3*g^6*h^3*x + 3*a*c^2*g^4*h^5*x + 3*a^2*c*g^2*h^7*x + a^3*h^9*x + c^3*g^7*h^2 + 3*a*c^2*g^5*h^4 + 3*a^2
*c*g^3*h^6 + a^3*g*h^8) - 1/8*(c*x^2 + a)^(5/2)*c^2*f*g^4/(c^3*g^6*h^3*x^2 + 3*a*c^2*g^4*h^5*x^2 + 3*a^2*c*g^2
*h^7*x^2 + a^3*h^9*x^2 + 2*c^3*g^7*h^2*x + 6*a*c^2*g^5*h^4*x + 6*a^2*c*g^3*h^6*x + 2*a^3*g*h^8*x + c^3*g^8*h +
3*a*c^2*g^6*h^3 + 3*a^2*c*g^4*h^5 + a^3*g^2*h^7) + 1/8*(c*x^2 + a)^(3/2)*c^3*f*g^4/(c^3*g^6*h^3 + 3*a*c^2*g^4
*h^5 + 3*a^2*c*g^2*h^7 + a^3*h^9) - 3/8*sqrt(c*x^2 + a)*c^4*d*g^3*x/(c^3*g^6*h^2 + 3*a*c^2*g^4*h^4 + 3*a^2*c*g
^2*h^6 + a^3*h^8) + 1/8*(c*x^2 + a)^(3/2)*c^3*d*g^3/(c^3*g^6*h^2*x + 3*a*c^2*g^4*h^4*x + 3*a^2*c*g^2*h^6*x + a
^3*h^8*x + c^3*g^7*h + 3*a*c^2*g^5*h^3 + 3*a^2*c*g^3*h^5 + a^3*g*h^7) + 1/8*(c*x^2 + a)^(5/2)*c^2*e*g^3/(c^3*g
^6*h^2*x^2 + 3*a*c^2*g^4*h^4*x^2 + 3*a^2*c*g^2*h^6*x^2 + a^3*h^8*x^2 + 2*c^3*g^7*h*x + 6*a*c^2*g^5*h^3*x + 6*a
^2*c*g^3*h^5*x + 2*a^3*g*h^7*x + c^3*g^8 + 3*a*c^2*g^6*h^2 + 3*a^2*c*g^4*h^4 + a^3*g^2*h^6) - 1/8*(c*x^2 + a)^
(3/2)*c^3*e*g^3/(c^3*g^6*h^2 + 3*a*c^2*g^4*h^4 + 3*a^2*c*g^2*h^6 + a^3*h^8) - 7/4*sqrt(c*x^2 + a)*c^3*f*g^4/(c
^2*g^4*h^5 + 2*a*c*g^2*h^7 + a^2*h^9) + 11/8*sqrt(c*x^2 + a)*c^3*f*g^3*x/(c^2*g^4*h^4 + 2*a*c*g^2*h^6 + a^2*h^
8) - 1/8*(c*x^2 + a)^(5/2)*c^2*d*g^2/(c^3*g^6*h*x^2 + 3*a*c^2*g^4*h^3*x^2 + 3*a^2*c*g^2*h^5*x^2 + a^3*h^7*x^2
+ 2*c^3*g^7*x + 6*a*c^2*g^5*h^2*x + 6*a^2*c*g^3*h^4*x + 2*a^3*g*h^6*x + c^3*g^8/h + 3*a*c^2*g^6*h + 3*a^2*c*g^
4*h^3 + a^3*g^2*h^5) + 1/8*(c*x^2 + a)^(3/2)*c^3*d*g^2/(c^3*g^6*h + 3*a*c^2*g^4*h^3 + 3*a^2*c*g^2*h^5 + a^3*h^
7) + 5/4*sqrt(c*x^2 + a)*c^3*e*g^3/(c^2*g^4*h^4 + 2*a*c*g^2*h^6 + a^2*h^8) - 1/4*(c*x^2 + a)^(5/2)*c*f*g^3/(c^
2*g^4*h^4*x^3 + 2*a*c*g^2*h^6*x^3 + a^2*h^8*x^3 + 3*c^2*g^5*h^3*x^2 + 6*a*c*g^3*h^5*x^2 + 3*a^2*g*h^7*x^2 + 3*
c^2*g^6*h^2*x + 6*a*c*g^4*h^4*x + 3*a^2*g^2*h^6*x + c^2*g^7*h + 2*a*c*g^5*h^3 + a^2*g^3*h^5) - 17/24*(c*x^2 +
a)^(3/2)*c^2*f*g^3/(c^2*g^4*h^4*x + 2*a*c*g^2*h^6*x + a^2*h^8*x + c^2*g^5*h^3 + 2*a*c*g^3*h^5 + a^2*g*h^7) - 7
/8*sqrt(c*x^2 + a)*c^3*e*g^2*x/(c^2*g^4*h^3 + 2*a*c*g^2*h^5 + a^2*h^7) - 3/4*sqrt(c*x^2 + a)*c^3*d*g^2/(c^2*g^
4*h^3 + 2*a*c*g^2*h^5 + a^2*h^7) + 1/4*(c*x^2 + a)^(5/2)*c*e*g^2/(c^2*g^4*h^3*x^3 + 2*a*c*g^2*h^5*x^3 + a^2*h^
7*x^3 + 3*c^2*g^5*h^2*x^2 + 6*a*c*g^3*h^4*x^2 + 3*a^2*g*h^6*x^2 + 3*c^2*g^6*h*x + 6*a*c*g^4*h^3*x + 3*a^2*g^2*
h^5*x + c^2*g^7 + 2*a*c*g^5*h^2 + a^2*g^3*h^4) + 13/24*(c*x^2 + a)^(3/2)*c^2*e*g^2/(c^2*g^4*h^3*x + 2*a*c*g^2*
h^5*x + a^2*h^7*x + c^2*g^5*h^2 + 2*a*c*g^3*h^4 + a^2*g*h^6) + 5/24*(c*x^2 + a)^(5/2)*c*f*g^2/(c^2*g^4*h^3*x^2
+ 2*a*c*g^2*h^5*x^2 + a^2*h^7*x^2 + 2*c^2*g^5*h^2*x + 4*a*c*g^3*h^4*x + 2*a^2*g*h^6*x + c^2*g^6*h + 2*a*c*g^4
*h^3 + a^2*g^2*h^5) - 5/24*(c*x^2 + a)^(3/2)*c^2*f*g^2/(c^2*g^4*h^3 + 2*a*c*g^2*h^5 + a^2*h^7) + 3/8*sqrt(c*x^
2 + a)*c^3*d*g*x/(c^2*g^4*h^2 + 2*a*c*g^2*h^4 + a^2*h^6) - 1/4*(c*x^2 + a)^(5/2)*c*d*g/(c^2*g^4*h^2*x^3 + 2*a*
c*g^2*h^4*x^3 + a^2*h^6*x^3 + 3*c^2*g^5*h*x^2 + 6*a*c*g^3*h^3*x^2 + 3*a^2*g*h^5*x^2 + 3*c^2*g^6*x + 6*a*c*g^4*
h^2*x + 3*a^2*g^2*h^4*x + c^2*g^7/h + 2*a*c*g^5*h + a^2*g^3*h^3) - 3/8*(c*x^2 + a)^(3/2)*c^2*d*g/(c^2*g^4*h^2*
x + 2*a*c*g^2*h^4*x + a^2*h^6*x + c^2*g^5*h + 2*a*c*g^3*h^3 + a^2*g*h^5) - 1/24*(c*x^2 + a)^(5/2)*c*e*g/(c^2*g
^4*h^2*x^2 + 2*a*c*g^2*h^4*x^2 + a^2*h^6*x^2 + 2*c^2*g^5*h*x + 4*a*c*g^3*h^3*x + 2*a^2*g*h^5*x + c^2*g^6 + 2*a
*c*g^4*h^2 + a^2*g^2*h^4) + 1/24*(c*x^2 + a)^(3/2)*c^2*e*g/(c^2*g^4*h^2 + 2*a*c*g^2*h^4 + a^2*h^6) - 1/4*(c*x^
2 + a)^(5/2)*f*g^2/(c*g^2*h^5*x^4 + a*h^7*x^4 + 4*c*g^3*h^4*x^3 + 4*a*g*h^6*x^3 + 6*c*g^4*h^3*x^2 + 6*a*g^2*h^
5*x^2 + 4*c*g^5*h^2*x + 4*a*g^3*h^4*x + c*g^6*h + a*g^4*h^3) + 39/8*sqrt(c*x^2 + a)*c^2*f*g^2/(c*g^2*h^5 + a*h
^7) - 7/2*sqrt(c*x^2 + a)*c^2*f*g*x/(c*g^2*h^4 + a*h^6) - 1/8*(c*x^2 + a)^(5/2)*c*d/(c^2*g^4*h*x^2 + 2*a*c*g^2
*h^3*x^2 + a^2*h^5*x^2 + 2*c^2*g^5*x + 4*a*c*g^3*h^2*x + 2*a^2*g*h^4*x + c^2*g^6/h + 2*a*c*g^4*h + a^2*g^2*h^3
) + 1/8*(c*x^2 + a)^(3/2)*c^2*d/(c^2*g^4*h + 2*a*c*g^2*h^3 + a^2*h^5) + 1/4*(c*x^2 + a)^(5/2)*e*g/(c*g^2*h^4*x
^4 + a*h^6*x^4 + 4*c*g^3*h^3*x^3 + 4*a*g*h^5*x^3 + 6*c*g^4*h^2*x^2 + 6*a*g^2*h^4*x^2 + 4*c*g^5*h*x + 4*a*g^3*h
^3*x + c*g^6 + a*g^4*h^2) - 15/8*sqrt(c*x^2 + a)*c^2*e*g/(c*g^2*h^4 + a*h^6) + 2/3*(c*x^2 + a)^(5/2)*f*g/(c*g^
2*h^4*x^3 + a*h^6*x^3 + 3*c*g^3*h^3*x^2 + 3*a*g*h^5*x^2 + 3*c*g^4*h^2*x + 3*a*g^2*h^4*x + c*g^5*h + a*g^3*h^3)
+ 11/6*(c*x^2 + a)^(3/2)*c*f*g/(c*g^2*h^4*x + a*h^6*x + c*g^3*h^3 + a*g*h^5) + sqrt(c*x^2 + a)*c^2*e*x/(c*g^2
*h^3 + a*h^5) - 1/4*(c*x^2 + a)^(5/2)*d/(c*g^2*h^3*x^4 + a*h^5*x^4 + 4*c*g^3*h^2*x^3 + 4*a*g*h^4*x^3 + 6*c*g^4
*h*x^2 + 6*a*g^2*h^3*x^2 + 4*c*g^5*x + 4*a*g^3*h^2*x + c*g^6/h + a*g^4*h) + 3/8*sqrt(c*x^2 + a)*c^2*d/(c*g^2*h
^3 + a*h^5) - 1/3*(c*x^2 + a)^(5/2)*e/(c*g^2*h^3*x^3 + a*h^5*x^3 + 3*c*g^3*h^2*x^2 + 3*a*g*h^4*x^2 + 3*c*g^4*h
*x + 3*a*g^2*h^3*x + c*g^5 + a*g^3*h^2) - 2/3*(c*x^2 + a)^(3/2)*c*e/(c*g^2*h^3*x + a*h^5*x + c*g^3*h^2 + a*g*h
^4) - 1/2*(c*x^2 + a)^(5/2)*f/(c*g^2*h^3*x^2 + a*h^5*x^2 + 2*c*g^3*h^2*x + 2*a*g*h^4*x + c*g^4*h + a*g^2*h^3)
+ 1/2*(c*x^2 + a)^(3/2)*c*f/(c*g^2*h^3 + a*h^5) - 5*c^(3/2)*f*g*arcsinh(c*x/sqrt(a*c))/h^6 + c^(3/2)*e*arcsinh
(c*x/sqrt(a*c))/h^5 + 3/8*c^4*f*g^6*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a
+ c*g^2/h^2)^(5/2)*h^11) - 3/8*c^4*e*g^5*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)
))/((a + c*g^2/h^2)^(5/2)*h^10) + 3/8*c^4*d*g^4*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*
x + g)))/((a + c*g^2/h^2)^(5/2)*h^9) - 7/4*c^3*f*g^4*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*a
bs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^9) + 5/4*c^3*e*g^3*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a
*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^8) - 3/4*c^3*d*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(s
qrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^7) + 39/8*c^2*f*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) -
a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^7) - 15/8*c^2*e*g*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g))
- a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^6) + 3/8*c^2*d*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) -
a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^5) + 3/2*sqrt(a + c*g^2/h^2)*c*f*arcsinh(c*g*x/(sqrt(a*c
)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^5 + 3/2*sqrt(c*x^2 + a)*c*f/h^5

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^5} \,d x \]

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)^5,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{5}}\, dx \]

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)

________________________________________________________________________________________