Integrand size = 21, antiderivative size = 74 \[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\frac {2 x \sqrt {1+\frac {c x}{b}} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},-\frac {c x}{b},-\frac {e x}{d}\right )}{\sqrt {b x+c x^2}} \] Output:
2*x*(1+c*x/b)^(1/2)*(e*x+d)^m*AppellF1(1/2,1/2,-m,3/2,-c*x/b,-e*x/d)/((1+e *x/d)^m)/(c*x^2+b*x)^(1/2)
Time = 0.77 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\frac {2 x \sqrt {\frac {b+c x}{b}} (d+e x)^m \left (\frac {d+e x}{d}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},-\frac {c x}{b},-\frac {e x}{d}\right )}{\sqrt {x (b+c x)}} \] Input:
Integrate[(d + e*x)^m/Sqrt[b*x + c*x^2],x]
Output:
(2*x*Sqrt[(b + c*x)/b]*(d + e*x)^m*AppellF1[1/2, 1/2, -m, 3/2, -((c*x)/b), -((e*x)/d)])/(Sqrt[x*(b + c*x)]*((d + e*x)/d)^m)
Time = 0.37 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.42, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1179, 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.
\(\displaystyle \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1179 |
\(\displaystyle \frac {\sqrt {-\frac {e x}{d}} \sqrt {1-\frac {c (d+e x)}{c d-b e}} \int \frac {(d+e x)^m}{\sqrt {1-\frac {d+e x}{d}} \sqrt {1-\frac {c (d+e x)}{c d-b e}}}d(d+e x)}{e \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\sqrt {-\frac {e x}{d}} (d+e x)^{m+1} \sqrt {1-\frac {c (d+e x)}{c d-b e}} \operatorname {AppellF1}\left (m+1,\frac {1}{2},\frac {1}{2},m+2,\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt {b x+c x^2}}\) |
Input:
Int[(d + e*x)^m/Sqrt[b*x + c*x^2],x]
Output:
(Sqrt[-((e*x)/d)]*(d + e*x)^(1 + m)*Sqrt[1 - (c*(d + e*x))/(c*d - b*e)]*Ap pellF1[1 + m, 1/2, 1/2, 2 + m, (d + e*x)/d, (c*(d + e*x))/(c*d - b*e)])/(e *(1 + m)*Sqrt[b*x + c*x^2])
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]
\[\int \frac {\left (e x +d \right )^{m}}{\sqrt {c \,x^{2}+b x}}d x\]
Input:
int((e*x+d)^m/(c*x^2+b*x)^(1/2),x)
Output:
int((e*x+d)^m/(c*x^2+b*x)^(1/2),x)
\[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x}} \,d x } \] Input:
integrate((e*x+d)^m/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
Output:
integral((e*x + d)^m/sqrt(c*x^2 + b*x), x)
\[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\sqrt {x \left (b + c x\right )}}\, dx \] Input:
integrate((e*x+d)**m/(c*x**2+b*x)**(1/2),x)
Output:
Integral((d + e*x)**m/sqrt(x*(b + c*x)), x)
\[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x}} \,d x } \] Input:
integrate((e*x+d)^m/(c*x^2+b*x)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x + d)^m/sqrt(c*x^2 + b*x), x)
\[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x}} \,d x } \] Input:
integrate((e*x+d)^m/(c*x^2+b*x)^(1/2),x, algorithm="giac")
Output:
integrate((e*x + d)^m/sqrt(c*x^2 + b*x), x)
Timed out. \[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{\sqrt {c\,x^2+b\,x}} \,d x \] Input:
int((d + e*x)^m/(b*x + c*x^2)^(1/2),x)
Output:
int((d + e*x)^m/(b*x + c*x^2)^(1/2), x)
\[ \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (e x +d \right )^{m}}{\sqrt {x}\, \sqrt {c x +b}}d x \] Input:
int((e*x+d)^m/(c*x^2+b*x)^(1/2),x)
Output:
int((d + e*x)**m/(sqrt(x)*sqrt(b + c*x)),x)