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

Optimal. Leaf size=488 \[ -\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (2 a^2 f h^4+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )+2 c^2 g^2 \left (10 f g^2-3 h (2 e g-d h)\right )\right )}{2 h^6 \sqrt {a h^2+c g^2}}+\frac {\sqrt {a+c x^2} \left (2 a^2 f h^4-c h x \left (a h^2 (7 f g-3 e h)+c g \left (10 f g^2-3 h (2 e g-d h)\right )\right )+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )+2 c^2 g^2 \left (10 f g^2-3 h (2 e g-d h)\right )\right )}{2 h^5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (2 \left (c g \left (-3 d h+6 e g-\frac {10 f g^2}{h}\right )-a h (7 f g-3 e h)\right )-x \left (2 a f h^2+c \left (5 f g^2-3 h (e g-d h)\right )\right )\right )}{6 h^2 (g+h x) \left (a h^2+c g^2\right )}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a h^2 (3 f g-e h)+2 c g \left (10 f g^2-3 h (2 e g-d h)\right )\right )}{2 h^6} \]

[Out]

-1/6*(2*c*g*(6*e*g-10*f*g^2/h-3*d*h)-2*a*h*(-3*e*h+7*f*g)-(2*a*f*h^2+c*(5*f*g^2-3*h*(-d*h+e*g)))*x)*(c*x^2+a)^
(3/2)/h^2/(a*h^2+c*g^2)/(h*x+g)-1/2*(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)^2-1/2*(3*a*h^2
*(-e*h+3*f*g)+2*c*g*(10*f*g^2-3*h*(-d*h+2*e*g)))*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))*c^(1/2)/h^6-1/2*(2*a^2*f*h
^4+2*c^2*g^2*(10*f*g^2-3*h*(-d*h+2*e*g))+a*c*h^2*(19*f*g^2-3*h*(-d*h+3*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)^(1/2)+1/2*(2*a^2*f*h^4+2*c^2*g^2*(10*f*g^2-3*h*(-d*h+2*e*g))+a*c*h
^2*(19*f*g^2-3*h*(-d*h+3*e*g))-c*h*(a*h^2*(-3*e*h+7*f*g)+c*g*(10*f*g^2-3*h*(-d*h+2*e*g)))*x)*(c*x^2+a)^(1/2)/h
^5/(a*h^2+c*g^2)

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Rubi [A]  time = 0.92, antiderivative size = 480, normalized size of antiderivative = 0.98, 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, 813, 815, 844, 217, 206, 725} \[ \frac {\sqrt {a+c x^2} \left (2 a^2 f h^3-c x \left (a h^2 (7 f g-3 e h)-3 c g h (2 e g-d h)+10 c f g^3\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-2 c^2 g^2 \left (-3 d h+6 e g-\frac {10 f g^2}{h}\right )\right )}{2 h^4 \left (a h^2+c g^2\right )}-\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (2 a^2 f h^4+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )\right )}{2 h^6 \sqrt {a h^2+c g^2}}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (2 \left (c g \left (-3 d h+6 e g-\frac {10 f g^2}{h}\right )-a h (7 f g-3 e h)\right )-x \left (2 a f h^2-3 c h (e g-d h)+5 c f g^2\right )\right )}{6 h^2 (g+h x) \left (a h^2+c g^2\right )}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a h^2 (3 f g-e h)-6 c g h (2 e g-d h)+20 c f g^3\right )}{2 h^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^2*f*h^3 - 2*c^2*g^2*(6*e*g - (10*f*g^2)/h - 3*d*h) + a*c*h*(19*f*g^2 - 3*h*(3*e*g - d*h)) - c*(10*c*f*g^
3 - 3*c*g*h*(2*e*g - d*h) + a*h^2*(7*f*g - 3*e*h))*x)*Sqrt[a + c*x^2])/(2*h^4*(c*g^2 + a*h^2)) - ((2*(c*g*(6*e
*g - (10*f*g^2)/h - 3*d*h) - a*h*(7*f*g - 3*e*h)) - (5*c*f*g^2 + 2*a*f*h^2 - 3*c*h*(e*g - d*h))*x)*(a + c*x^2)
^(3/2))/(6*h^2*(c*g^2 + a*h^2)*(g + h*x)) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(2*h*(c*g^2 + a*h^2)*(
g + h*x)^2) - (Sqrt[c]*(20*c*f*g^3 - 6*c*g*h*(2*e*g - d*h) + 3*a*h^2*(3*f*g - e*h))*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(2*h^6) - ((2*a^2*f*h^4 + 2*c^2*(10*f*g^4 - 3*g^2*h*(2*e*g - d*h)) + a*c*h^2*(19*f*g^2 - 3*h*(3*e*
g - d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(2*h^6*Sqrt[c*g^2 + a*h^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 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 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 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)^3} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\int \frac {\left (-2 (c d g-a f g+a e h)-\left (2 a f h-c \left (3 e g-\frac {5 f g^2}{h}-3 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^2} \, dx}{2 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (2 \left (c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\int \frac {\left (2 a \left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right )-\frac {4 c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x}{h}\right ) \sqrt {a+c x^2}}{g+h x} \, dx}{4 h^2 \left (c g^2+a h^2\right )}\\ &=\frac {\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac {\left (2 \left (c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\int \frac {4 a c \left (c g^2+a h^2\right ) \left (10 c f g^2+2 a f h^2-3 c h (2 e g-d h)\right )-\frac {4 c^2 \left (c g^2+a h^2\right ) \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e h)\right ) x}{h}}{(g+h x) \sqrt {a+c x^2}} \, dx}{8 c h^4 \left (c g^2+a h^2\right )}\\ &=\frac {\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac {\left (2 \left (c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (c \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e h)\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 h^6}+\frac {\left (2 a^2 f h^4+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{2 h^6}\\ &=\frac {\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac {\left (2 \left (c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (c \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e h)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 h^6}-\frac {\left (2 a^2 f h^4+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\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 )}{2 h^6}\\ &=\frac {\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac {\left (2 \left (c g \left (6 e g-\frac {10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\sqrt {c} \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e h)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 h^6}-\frac {\left (2 a^2 f h^4+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )+a c h^2 \left (19 f g^2-3 h (3 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 )}{2 h^6 \sqrt {c g^2+a h^2}}\\ \end {align*}

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Mathematica [A]  time = 0.65, size = 435, normalized size = 0.89 \[ \frac {-\frac {3 \log \left (\sqrt {a+c x^2} \sqrt {a h^2+c g^2}+a h-c g x\right ) \left (2 a^2 f h^4+a c h^2 \left (3 h (d h-3 e g)+19 f g^2\right )+2 c^2 \left (3 g^2 h (d h-2 e g)+10 f g^4\right )\right )}{\sqrt {a h^2+c g^2}}+\frac {3 \log (g+h x) \left (2 a^2 f h^4+a c h^2 \left (3 h (d h-3 e g)+19 f g^2\right )+2 c^2 \left (3 g^2 h (d h-2 e g)+10 f g^4\right )\right )}{\sqrt {a h^2+c g^2}}-3 \sqrt {c} \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right ) \left (-3 a h^2 (e h-3 f g)+6 c g h (d h-2 e g)+20 c f g^3\right )+\frac {h \sqrt {a+c x^2} \left (a h^2 \left (f \left (17 g^2+28 g h x+8 h^2 x^2\right )-3 h (d h+e g+2 e h x)\right )+c \left (3 h \left (d h \left (6 g^2+9 g h x+2 h^2 x^2\right )+e \left (-12 g^3-18 g^2 h x-4 g h^2 x^2+h^3 x^3\right )\right )+f \left (60 g^4+90 g^3 h x+20 g^2 h^2 x^2-5 g h^3 x^3+2 h^4 x^4\right )\right )\right )}{(g+h x)^2}}{6 h^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((h*Sqrt[a + c*x^2]*(a*h^2*(-3*h*(e*g + d*h + 2*e*h*x) + f*(17*g^2 + 28*g*h*x + 8*h^2*x^2)) + c*(f*(60*g^4 + 9
0*g^3*h*x + 20*g^2*h^2*x^2 - 5*g*h^3*x^3 + 2*h^4*x^4) + 3*h*(d*h*(6*g^2 + 9*g*h*x + 2*h^2*x^2) + e*(-12*g^3 -
18*g^2*h*x - 4*g*h^2*x^2 + h^3*x^3)))))/(g + h*x)^2 + (3*(2*a^2*f*h^4 + a*c*h^2*(19*f*g^2 + 3*h*(-3*e*g + d*h)
) + 2*c^2*(10*f*g^4 + 3*g^2*h*(-2*e*g + d*h)))*Log[g + h*x])/Sqrt[c*g^2 + a*h^2] - 3*Sqrt[c]*(20*c*f*g^3 + 6*c
*g*h*(-2*e*g + d*h) - 3*a*h^2*(-3*f*g + e*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (3*(2*a^2*f*h^4 + a*c*h^2*(
19*f*g^2 + 3*h*(-3*e*g + d*h)) + 2*c^2*(10*f*g^4 + 3*g^2*h*(-2*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h
^2]*Sqrt[a + c*x^2]])/Sqrt[c*g^2 + a*h^2])/(6*h^6)

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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)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.43, size = 1036, normalized size = 2.12 \[ \frac {1}{6} \, \sqrt {c x^{2} + a} {\left (x {\left (\frac {2 \, c f x}{h^{3}} - \frac {3 \, {\left (3 \, c^{2} f g h^{14} - c^{2} h^{15} e\right )}}{c h^{18}}\right )} + \frac {2 \, {\left (18 \, c^{2} f g^{2} h^{13} + 3 \, c^{2} d h^{15} + 4 \, a c f h^{15} - 9 \, c^{2} g h^{14} e\right )}}{c h^{18}}\right )} + \frac {{\left (20 \, c^{\frac {3}{2}} f g^{3} + 6 \, c^{\frac {3}{2}} d g h^{2} + 9 \, a \sqrt {c} f g h^{2} - 12 \, c^{\frac {3}{2}} g^{2} h e - 3 \, a \sqrt {c} h^{3} e\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, h^{6}} + \frac {{\left (20 \, c^{2} f g^{4} + 6 \, c^{2} d g^{2} h^{2} + 19 \, a c f g^{2} h^{2} + 3 \, a c d h^{4} + 2 \, a^{2} f h^{4} - 12 \, c^{2} g^{3} h e - 9 \, a c g h^{3} 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 {10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} f g^{4} h + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d g^{2} h^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c f g^{2} h^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c d h^{5} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} g^{3} h^{2} e - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c g h^{4} e + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} f g^{5} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d g^{3} h^{2} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} f g^{3} h^{2} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d g h^{4} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} \sqrt {c} f g h^{4} - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} g^{4} h e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} g^{2} h^{3} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} \sqrt {c} h^{5} e - 26 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} f g^{4} h - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d g^{2} h^{3} - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c f g^{2} h^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c d h^{5} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} g^{3} h^{2} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c g h^{4} e + 9 \, a^{2} c^{\frac {3}{2}} f g^{3} h^{2} + 5 \, a^{2} c^{\frac {3}{2}} d g h^{4} + 4 \, a^{3} \sqrt {c} f g h^{4} - 7 \, a^{2} c^{\frac {3}{2}} g^{2} h^{3} e - 2 \, a^{3} \sqrt {c} h^{5} e}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} h + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} g - a h\right )}^{2} h^{6}} \]

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)^3,x, algorithm="giac")

[Out]

1/6*sqrt(c*x^2 + a)*(x*(2*c*f*x/h^3 - 3*(3*c^2*f*g*h^14 - c^2*h^15*e)/(c*h^18)) + 2*(18*c^2*f*g^2*h^13 + 3*c^2
*d*h^15 + 4*a*c*f*h^15 - 9*c^2*g*h^14*e)/(c*h^18)) + 1/2*(20*c^(3/2)*f*g^3 + 6*c^(3/2)*d*g*h^2 + 9*a*sqrt(c)*f
*g*h^2 - 12*c^(3/2)*g^2*h*e - 3*a*sqrt(c)*h^3*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/h^6 + (20*c^2*f*g^4 +
6*c^2*d*g^2*h^2 + 19*a*c*f*g^2*h^2 + 3*a*c*d*h^4 + 2*a^2*f*h^4 - 12*c^2*g^3*h*e - 9*a*c*g*h^3*e)*arctan(-((sqr
t(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) + (10*(sqrt(c)*x - s
qrt(c*x^2 + a))^3*c^2*f*g^4*h + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d*g^2*h^3 + 5*(sqrt(c)*x - sqrt(c*x^2 +
a))^3*a*c*f*g^2*h^3 + (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*d*h^5 - 8*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*g^3*h^
2*e - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*g*h^4*e + 18*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*f*g^5 + 10*(s
qrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*d*g^3*h^2 - (sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*f*g^3*h^2 - 5*(sqr
t(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*g*h^4 - 4*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*sqrt(c)*f*g*h^4 - 14*(sq
rt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*g^4*h*e + 3*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*g^2*h^3*e + 2*(sqrt
(c)*x - sqrt(c*x^2 + a))^2*a^2*sqrt(c)*h^5*e - 26*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*f*g^4*h - 14*(sqrt(c)*x
- sqrt(c*x^2 + a))*a*c^2*d*g^2*h^3 - 11*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*f*g^2*h^3 + (sqrt(c)*x - sqrt(c*x^
2 + a))*a^2*c*d*h^5 + 20*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*g^3*h^2*e + 5*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c
*g*h^4*e + 9*a^2*c^(3/2)*f*g^3*h^2 + 5*a^2*c^(3/2)*d*g*h^4 + 4*a^3*sqrt(c)*f*g*h^4 - 7*a^2*c^(3/2)*g^2*h^3*e -
2*a^3*sqrt(c)*h^5*e)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g - a*h)^2
*h^6)

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maple [B]  time = 0.02, size = 7817, normalized size = 16.02 \[ \text {output too large to display} \]

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)^3,x)

[Out]

result too large to display

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maxima [B]  time = 0.88, size = 1299, normalized size = 2.66 \[ \text {result too large to display} \]

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)^3,x, algorithm="maxima")

[Out]

3/2*sqrt(c*x^2 + a)*c^2*f*g^4/(c*g^2*h^5 + a*h^7) - 3/2*sqrt(c*x^2 + a)*c^2*f*g^3*x/(c*g^2*h^4 + a*h^6) - 3/2*
sqrt(c*x^2 + a)*c^2*e*g^3/(c*g^2*h^4 + a*h^6) + 1/2*(c*x^2 + a)^(3/2)*c*f*g^3/(c*g^2*h^4*x + a*h^6*x + c*g^3*h
^3 + a*g*h^5) + 3/2*sqrt(c*x^2 + a)*c^2*e*g^2*x/(c*g^2*h^3 + a*h^5) + 3/2*sqrt(c*x^2 + a)*c^2*d*g^2/(c*g^2*h^3
+ a*h^5) - 1/2*(c*x^2 + a)^(3/2)*c*e*g^2/(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*g^2/(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*g^2/(c*g^2*h^3 + a*h^5) - 3/2*sqrt(c*x^2 + a)*c^2*d*g*x/(c*g^2*h^2 + a*h^4) + 1/2*(c*x^2 + a)^(3/2)*c*d*
g/(c*g^2*h^2*x + a*h^4*x + c*g^3*h + a*g*h^3) + 1/2*(c*x^2 + a)^(5/2)*e*g/(c*g^2*h^2*x^2 + a*h^4*x^2 + 2*c*g^3
*h*x + 2*a*g*h^3*x + c*g^4 + a*g^2*h^2) - 1/2*(c*x^2 + a)^(3/2)*c*e*g/(c*g^2*h^2 + a*h^4) - 1/2*(c*x^2 + a)^(5
/2)*d/(c*g^2*h*x^2 + a*h^3*x^2 + 2*c*g^3*x + 2*a*g*h^2*x + c*g^4/h + a*g^2*h) + 1/2*(c*x^2 + a)^(3/2)*c*d/(c*g
^2*h + a*h^3) + 2*(c*x^2 + a)^(3/2)*f*g/(h^4*x + g*h^3) - (c*x^2 + a)^(3/2)*e/(h^3*x + g*h^2) - 7/2*sqrt(c*x^2
+ a)*c*f*g*x/h^4 + 3/2*sqrt(c*x^2 + a)*c*e*x/h^3 - 10*c^(3/2)*f*g^3*arcsinh(c*x/sqrt(a*c))/h^6 + 6*c^(3/2)*e*
g^2*arcsinh(c*x/sqrt(a*c))/h^5 - 3*c^(3/2)*d*g*arcsinh(c*x/sqrt(a*c))/h^4 - 9/2*a*sqrt(c)*f*g*arcsinh(c*x/sqrt
(a*c))/h^4 + 3/2*a*sqrt(c)*e*arcsinh(c*x/sqrt(a*c))/h^3 + 3/2*c^2*f*g^4*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) - 3/2*c^2*e*g^3*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) + 3/2*c^2*d*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^5) + 15/2*sqrt(a + c*g^2/h^2)*c*f*g^2*arcsinh(c*
g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^5 - 9/2*sqrt(a + c*g^2/h^2)*c*e*g*arcsinh(c*g*x
/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^4 + 3/2*sqrt(a + c*g^2/h^2)*c*d*arcsinh(c*g*x/(sqr
t(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 + (a + c*g^2/h^2)^(3/2)*f*arcsinh(c*g*x/(sqrt(a*c)*ab
s(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 + 17/2*sqrt(c*x^2 + a)*c*f*g^2/h^5 - 9/2*sqrt(c*x^2 + a)*c*e*g
/h^4 + 3/2*sqrt(c*x^2 + a)*c*d/h^3 + 1/3*(c*x^2 + a)^(3/2)*f/h^3 + sqrt(c*x^2 + a)*a*f/h^3

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

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)^3,x)

[Out]

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

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

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)**3,x)

[Out]

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

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