Integrand size = 20, antiderivative size = 77 \[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\frac {\sqrt {-1-x} (4+3 x)^{1+m} \sqrt {6+3 x} \operatorname {AppellF1}\left (1+m,\frac {1}{2},\frac {1}{2},2+m,4+3 x,\frac {1}{2} (-4-3 x)\right )}{\sqrt {6} (1+m) \sqrt {2+3 x+x^2}} \] Output:
1/6*(-1-x)^(1/2)*(4+3*x)^(1+m)*(6+3*x)^(1/2)*AppellF1(1+m,1/2,1/2,2+m,-2-3 /2*x,4+3*x)*6^(1/2)/(1+m)/(x^2+3*x+2)^(1/2)
Time = 0.35 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\frac {\sqrt {-((1+x) (2+x))} (4+3 x)^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{2},\frac {1}{2},2+m,-2-\frac {3 x}{2},4+3 x\right )}{\sqrt {2} (1+m) \sqrt {2+3 x+x^2}} \] Input:
Integrate[(4 + 3*x)^m/Sqrt[2 + 3*x + x^2],x]
Output:
(Sqrt[-((1 + x)*(2 + x))]*(4 + 3*x)^(1 + m)*AppellF1[1 + m, 1/2, 1/2, 2 + m, -2 - (3*x)/2, 4 + 3*x])/(Sqrt[2]*(1 + m)*Sqrt[2 + 3*x + x^2])
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1179, 27, 150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(3 x+4)^m}{\sqrt {x^2+3 x+2}} \, dx\) |
\(\Big \downarrow \) 1179 |
\(\displaystyle \frac {\sqrt {-x-1} \sqrt {3 x+6} \int \frac {\sqrt {2} (3 x+4)^m}{\sqrt {-3 x-3} \sqrt {3 x+6}}d(3 x+4)}{\sqrt {6} \sqrt {x^2+3 x+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {-x-1} \sqrt {3 x+6} \int \frac {(3 x+4)^m}{\sqrt {-3 x-3} \sqrt {3 x+6}}d(3 x+4)}{\sqrt {3} \sqrt {x^2+3 x+2}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\sqrt {-x-1} \sqrt {3 x+6} (3 x+4)^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2},\frac {1}{2},m+2,3 x+4,\frac {1}{2} (-3 x-4)\right )}{\sqrt {6} (m+1) \sqrt {x^2+3 x+2}}\) |
Input:
Int[(4 + 3*x)^m/Sqrt[2 + 3*x + x^2],x]
Output:
(Sqrt[-1 - x]*(4 + 3*x)^(1 + m)*Sqrt[6 + 3*x]*AppellF1[1 + m, 1/2, 1/2, 2 + m, 4 + 3*x, (-4 - 3*x)/2])/(Sqrt[6]*(1 + m)*Sqrt[2 + 3*x + x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> 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] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) ^p) Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d - e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m , p}, x]
\[\int \frac {\left (3 x +4\right )^{m}}{\sqrt {x^{2}+3 x +2}}d x\]
Input:
int((3*x+4)^m/(x^2+3*x+2)^(1/2),x)
Output:
int((3*x+4)^m/(x^2+3*x+2)^(1/2),x)
\[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\int { \frac {{\left (3 \, x + 4\right )}^{m}}{\sqrt {x^{2} + 3 \, x + 2}} \,d x } \] Input:
integrate((4+3*x)^m/(x^2+3*x+2)^(1/2),x, algorithm="fricas")
Output:
integral((3*x + 4)^m/sqrt(x^2 + 3*x + 2), x)
\[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\int \frac {\left (3 x + 4\right )^{m}}{\sqrt {\left (x + 1\right ) \left (x + 2\right )}}\, dx \] Input:
integrate((4+3*x)**m/(x**2+3*x+2)**(1/2),x)
Output:
Integral((3*x + 4)**m/sqrt((x + 1)*(x + 2)), x)
\[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\int { \frac {{\left (3 \, x + 4\right )}^{m}}{\sqrt {x^{2} + 3 \, x + 2}} \,d x } \] Input:
integrate((4+3*x)^m/(x^2+3*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate((3*x + 4)^m/sqrt(x^2 + 3*x + 2), x)
\[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\int { \frac {{\left (3 \, x + 4\right )}^{m}}{\sqrt {x^{2} + 3 \, x + 2}} \,d x } \] Input:
integrate((4+3*x)^m/(x^2+3*x+2)^(1/2),x, algorithm="giac")
Output:
integrate((3*x + 4)^m/sqrt(x^2 + 3*x + 2), x)
Timed out. \[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\int \frac {{\left (3\,x+4\right )}^m}{\sqrt {x^2+3\,x+2}} \,d x \] Input:
int((3*x + 4)^m/(3*x + x^2 + 2)^(1/2),x)
Output:
int((3*x + 4)^m/(3*x + x^2 + 2)^(1/2), x)
\[ \int \frac {(4+3 x)^m}{\sqrt {2+3 x+x^2}} \, dx=\int \frac {\left (3 x +4\right )^{m}}{\sqrt {x^{2}+3 x +2}}d x \] Input:
int((4+3*x)^m/(x^2+3*x+2)^(1/2),x)
Output:
int((3*x + 4)**m/sqrt(x**2 + 3*x + 2),x)