Integrand size = 46, antiderivative size = 112 \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{3 c d (d+e x)^{3/2}} \] Output:
2/3*(g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*hypergeom([3/2, -n], [5/2],-g*(c*d*x+a*e)/(-a*e*g+c*d*f))/c/d/(e*x+d)^(3/2)/((c*d*(g*x+f)/(-a*e *g+c*d*f))^n)
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.89 \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )}{3 c d (d+e x)^{3/2}} \] Input:
Integrate[((f + g*x)^n*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(f + g*x)^n*Hypergeometric2F1[3/2, -n, 5/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*c*d*(d + e*x)^(3/2)*((c*d*( f + g*x))/(c*d*f - a*e*g))^n)
Time = 0.43 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1268, 80, 79}
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.
| \(\displaystyle \int \frac {(f+g x)^n \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \int \sqrt {a e+c d x} (f+g x)^ndx}{\sqrt {d+e x} \sqrt {a e+c d x}}\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {(f+g x)^n \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \int \sqrt {a e+c d x} \left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^ndx}{\sqrt {d+e x} \sqrt {a e+c d x}}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {2 (f+g x)^n (a e+c d x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{3 c d \sqrt {d+e x}}\) |
Input:
Int[((f + g*x)^n*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x ],x]
Output:
(2*(a*e + c*d*x)*(f + g*x)^n*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]*H ypergeometric2F1[3/2, -n, 5/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))])/(3*c *d*Sqrt[d + e*x]*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]) Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
\[\int \frac {\left (g x +f \right )^{n} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{\sqrt {e x +d}}d x\]
Input:
int((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^(1/2),x)
Output:
int((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^(1/2),x)
\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}} \,d x } \] Input:
integrate((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2), x, algorithm="fricas")
Output:
integral(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^n/sqrt(e*x + d), x)
\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{n}}{\sqrt {d + e x}}\, dx \] Input:
integrate((g*x+f)**n*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)** (1/2),x)
Output:
Integral(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**n/sqrt(d + e*x), x)
\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}} \,d x } \] Input:
integrate((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2), x, algorithm="maxima")
Output:
integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^n/sqrt(e*x + d), x)
\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}} \,d x } \] Input:
integrate((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2), x, algorithm="giac")
Output:
integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^n/sqrt(e*x + d), x)
Timed out. \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\sqrt {d+e\,x}} \,d x \] Input:
int(((f + g*x)^n*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^ (1/2),x)
Output:
int(((f + g*x)^n*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^ (1/2), x)
\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\text {too large to display} \] Input:
int((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)
Output:
(2*(2*(f + g*x)**n*sqrt(a*e + c*d*x)*a*e*f*n + (f + g*x)**n*sqrt(a*e + c*d *x)*a*e*f + 2*(f + g*x)**n*sqrt(a*e + c*d*x)*a*e*g*n*x + (f + g*x)**n*sqrt (a*e + c*d*x)*c*d*f*x + 4*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(4*a**2*e **2*f*g*n**2 + 6*a**2*e**2*f*g*n + 4*a**2*e**2*g**2*n**2*x + 6*a**2*e**2*g **2*n*x + 2*a*c*d*e*f**2*n + 3*a*c*d*e*f**2 + 4*a*c*d*e*f*g*n**2*x + 8*a*c *d*e*f*g*n*x + 3*a*c*d*e*f*g*x + 4*a*c*d*e*g**2*n**2*x**2 + 6*a*c*d*e*g**2 *n*x**2 + 2*c**2*d**2*f**2*n*x + 3*c**2*d**2*f**2*x + 2*c**2*d**2*f*g*n*x* *2 + 3*c**2*d**2*f*g*x**2),x)*a**3*e**3*g**3*n**3 + 6*int(((f + g*x)**n*sq rt(a*e + c*d*x)*x)/(4*a**2*e**2*f*g*n**2 + 6*a**2*e**2*f*g*n + 4*a**2*e**2 *g**2*n**2*x + 6*a**2*e**2*g**2*n*x + 2*a*c*d*e*f**2*n + 3*a*c*d*e*f**2 + 4*a*c*d*e*f*g*n**2*x + 8*a*c*d*e*f*g*n*x + 3*a*c*d*e*f*g*x + 4*a*c*d*e*g** 2*n**2*x**2 + 6*a*c*d*e*g**2*n*x**2 + 2*c**2*d**2*f**2*n*x + 3*c**2*d**2*f **2*x + 2*c**2*d**2*f*g*n*x**2 + 3*c**2*d**2*f*g*x**2),x)*a**3*e**3*g**3*n **2 - 8*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(4*a**2*e**2*f*g*n**2 + 6*a **2*e**2*f*g*n + 4*a**2*e**2*g**2*n**2*x + 6*a**2*e**2*g**2*n*x + 2*a*c*d* e*f**2*n + 3*a*c*d*e*f**2 + 4*a*c*d*e*f*g*n**2*x + 8*a*c*d*e*f*g*n*x + 3*a *c*d*e*f*g*x + 4*a*c*d*e*g**2*n**2*x**2 + 6*a*c*d*e*g**2*n*x**2 + 2*c**2*d **2*f**2*n*x + 3*c**2*d**2*f**2*x + 2*c**2*d**2*f*g*n*x**2 + 3*c**2*d**2*f *g*x**2),x)*a**2*c*d*e**2*f*g**2*n**3 - 10*int(((f + g*x)**n*sqrt(a*e + c* d*x)*x)/(4*a**2*e**2*f*g*n**2 + 6*a**2*e**2*f*g*n + 4*a**2*e**2*g**2*n*...