\(\int (g+h x)^m \sqrt {a+c x^2} (d+e x+f x^2) \, dx\) [137]

Optimal result

Integrand size = 29, antiderivative size = 403 \[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac {\left (a f h^2 (1+m)-c \left (3 f g^2-h (e g-d h) (4+m)\right )\right ) (g+h x)^{1+m} \sqrt {a+c x^2} \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{c h^3 (1+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}-\frac {(3 f g-e h (4+m)) (g+h x)^{2+m} \sqrt {a+c x^2} \operatorname {AppellF1}\left (2+m,-\frac {1}{2},-\frac {1}{2},3+m,\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{h^3 (2+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}} \]

[Out]

f*(h*x+g)^(1+m)*(c*x^2+a)^(3/2)/c/h/(4+m)-(a*f*h^2*(1+m)-c*(3*f*g^2-h*(-d*h+e*g)*(4+m)))*(h*x+g)^(1+m)*AppellF
1(1+m,-1/2,-1/2,2+m,(h*x+g)/(g-h*(-a)^(1/2)/c^(1/2)),(h*x+g)/(g+h*(-a)^(1/2)/c^(1/2)))*(c*x^2+a)^(1/2)/c/h^3/(
1+m)/(4+m)/(1+(-h*x-g)/(g-h*(-a)^(1/2)/c^(1/2)))^(1/2)/(1+(-h*x-g)/(g+h*(-a)^(1/2)/c^(1/2)))^(1/2)-(3*f*g-e*h*
(4+m))*(h*x+g)^(2+m)*AppellF1(2+m,-1/2,-1/2,3+m,(h*x+g)/(g-h*(-a)^(1/2)/c^(1/2)),(h*x+g)/(g+h*(-a)^(1/2)/c^(1/
2)))*(c*x^2+a)^(1/2)/h^3/(2+m)/(4+m)/(1+(-h*x-g)/(g-h*(-a)^(1/2)/c^(1/2)))^(1/2)/(1+(-h*x-g)/(g+h*(-a)^(1/2)/c
^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1668, 858, 774, 138} \[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {a+c x^2} (g+h x)^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {1}{2},-\frac {1}{2},m+2,\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right ) \left (-a f h^2 (m+1)-c h (m+4) (e g-d h)+3 c f g^2\right )}{c h^3 (m+1) (m+4) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{\frac {\sqrt {-a} h}{\sqrt {c}}+g}}}-\frac {\sqrt {a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) \operatorname {AppellF1}\left (m+2,-\frac {1}{2},-\frac {1}{2},m+3,\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{h^3 (m+2) (m+4) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{\frac {\sqrt {-a} h}{\sqrt {c}}+g}}}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]

[In]

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

[Out]

(f*(g + h*x)^(1 + m)*(a + c*x^2)^(3/2))/(c*h*(4 + m)) + ((3*c*f*g^2 - a*f*h^2*(1 + m) - c*h*(e*g - d*h)*(4 + m
))*(g + h*x)^(1 + m)*Sqrt[a + c*x^2]*AppellF1[1 + m, -1/2, -1/2, 2 + m, (g + h*x)/(g - (Sqrt[-a]*h)/Sqrt[c]),
(g + h*x)/(g + (Sqrt[-a]*h)/Sqrt[c])])/(c*h^3*(1 + m)*(4 + m)*Sqrt[1 - (g + h*x)/(g - (Sqrt[-a]*h)/Sqrt[c])]*S
qrt[1 - (g + h*x)/(g + (Sqrt[-a]*h)/Sqrt[c])]) - ((3*f*g - e*h*(4 + m))*(g + h*x)^(2 + m)*Sqrt[a + c*x^2]*Appe
llF1[2 + m, -1/2, -1/2, 3 + m, (g + h*x)/(g - (Sqrt[-a]*h)/Sqrt[c]), (g + h*x)/(g + (Sqrt[-a]*h)/Sqrt[c])])/(h
^3*(2 + m)*(4 + m)*Sqrt[1 - (g + h*x)/(g - (Sqrt[-a]*h)/Sqrt[c])]*Sqrt[1 - (g + h*x)/(g + (Sqrt[-a]*h)/Sqrt[c]
)])

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 774

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

Rule 858

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 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\int (g+h x)^m \left (-h^2 (a f (1+m)-c d (4+m))-c h (3 f g-e h (4+m)) x\right ) \sqrt {a+c x^2} \, dx}{c h^2 (4+m)} \\ & = \frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac {(3 f g-e h (4+m)) \int (g+h x)^{1+m} \sqrt {a+c x^2} \, dx}{h^2 (4+m)}+\frac {\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) \int (g+h x)^m \sqrt {a+c x^2} \, dx}{c h^2 (4+m)} \\ & = \frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac {\left ((3 f g-e h (4+m)) \sqrt {a+c x^2}\right ) \text {Subst}\left (\int x^{1+m} \sqrt {1-\frac {x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}} \, dx,x,g+h x\right )}{h^3 (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}+\frac {\left (\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) \sqrt {a+c x^2}\right ) \text {Subst}\left (\int x^m \sqrt {1-\frac {x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}} \, dx,x,g+h x\right )}{c h^3 (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}} \\ & = \frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) (g+h x)^{1+m} \sqrt {a+c x^2} F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{c h^3 (1+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}-\frac {(3 f g-e h (4+m)) (g+h x)^{2+m} \sqrt {a+c x^2} F_1\left (2+m;-\frac {1}{2},-\frac {1}{2};3+m;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{h^3 (2+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}} \\ \end{align*}

Mathematica [F]

\[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (h x +g \right )^{m} \left (f \,x^{2}+e x +d \right ) \sqrt {c \,x^{2}+a}d x\]

[In]

int((h*x+g)^m*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x)

[Out]

int((h*x+g)^m*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x)

Fricas [F]

\[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int { \sqrt {c x^{2} + a} {\left (f x^{2} + e x + d\right )} {\left (h x + g\right )}^{m} \,d x } \]

[In]

integrate((h*x+g)^m*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + a)*(f*x^2 + e*x + d)*(h*x + g)^m, x)

Sympy [F]

\[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int \sqrt {a + c x^{2}} \left (g + h x\right )^{m} \left (d + e x + f x^{2}\right )\, dx \]

[In]

integrate((h*x+g)**m*(f*x**2+e*x+d)*(c*x**2+a)**(1/2),x)

[Out]

Integral(sqrt(a + c*x**2)*(g + h*x)**m*(d + e*x + f*x**2), x)

Maxima [F]

\[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int { \sqrt {c x^{2} + a} {\left (f x^{2} + e x + d\right )} {\left (h x + g\right )}^{m} \,d x } \]

[In]

integrate((h*x+g)^m*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)*(f*x^2 + e*x + d)*(h*x + g)^m, x)

Giac [F]

\[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int { \sqrt {c x^{2} + a} {\left (f x^{2} + e x + d\right )} {\left (h x + g\right )}^{m} \,d x } \]

[In]

integrate((h*x+g)^m*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + a)*(f*x^2 + e*x + d)*(h*x + g)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int {\left (g+h\,x\right )}^m\,\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right ) \,d x \]

[In]

int((g + h*x)^m*(a + c*x^2)^(1/2)*(d + e*x + f*x^2),x)

[Out]

int((g + h*x)^m*(a + c*x^2)^(1/2)*(d + e*x + f*x^2), x)