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

Optimal. Leaf size=403 \[ -\frac{\sqrt{a+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right ) \left (a f h^2 (m+1)-c \left (3 f g^2-h (m+4) (e g-d h)\right )\right )}{c h^3 (m+1) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}-\frac{\sqrt{a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (m+2) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}+\frac{f \left (a+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]

[Out]

(f*(g + h*x)^(1 + m)*(a + c*x^2)^(3/2))/(c*h*(4 + m)) - ((a*f*h^2*(1 + m) - c*(3*f*g^2 - 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]
)])

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Rubi [A]  time = 0.453214, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1654, 844, 760, 133} \[ \frac{\sqrt{a+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right ) \left (-a f h^2 (m+1)-c h (m+4) (e g-d h)+3 c f g^2\right )}{c h^3 (m+1) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}-\frac{\sqrt{a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (m+2) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}+\frac{f \left (a+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]

Antiderivative was successfully verified.

[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 1654

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]))

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 760

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 133

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

Rubi steps

\begin{align*} \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{\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 ) \operatorname{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 ) \operatorname{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]  time = 0.725941, size = 0, normalized size = 0. \[ \int (g+h x)^m \sqrt{a+c x^2} \left (d+e x+f x^2\right ) \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [F]  time = 0.742, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{m} \left ( f{x}^{2}+ex+d \right ) \sqrt{c{x}^{2}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{2} + a}{\left (f x^{2} + e x + d\right )}{\left (h x + g\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c x^{2} + a}{\left (f x^{2} + e x + d\right )}{\left (h x + g\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + c x^{2}} \left (g + h x\right )^{m} \left (d + e x + f x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{2} + a}{\left (f x^{2} + e x + d\right )}{\left (h x + g\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)