Integrand size = 46, antiderivative size = 112 \[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{5 c d (d+e x)^{5/2}} \] Output:
2/5*(g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*hypergeom([5/2, -n], [7/2],-g*(c*d*x+a*e)/(-a*e*g+c*d*f))/c/d/(e*x+d)^(5/2)/((c*d*(g*x+f)/(-a*e *g+c*d*f))^n)
Time = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.89 \[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )}{5 c d (d+e x)^{5/2}} \] Input:
Integrate[((f + g*x)^n*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(f + g*x)^n*Hypergeometric2F1[5/2, -n, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*c*d*(d + e*x)^(5/2)*((c*d*( f + g*x))/(c*d*f - a*e*g))^n)
Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.09, 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.
| \(\displaystyle \int \frac {(f+g x)^n \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \int (a e+c d x)^{3/2} (f+g x)^ndx}{\sqrt {d+e x} \sqrt {a e+c d x}}\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {(f+g x)^n \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \int (a e+c d x)^{3/2} \left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^ndx}{\sqrt {d+e x} \sqrt {a e+c d x}}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {2 (f+g x)^n (a e+c d x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{5 c d \sqrt {d+e x}}\) |
Input:
Int[((f + g*x)^n*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^ (3/2),x]
Output:
(2*(a*e + c*d*x)^2*(f + g*x)^n*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] *Hypergeometric2F1[5/2, -n, 7/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))])/(5 *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]
\[\int \frac {\left (g x +f \right )^{n} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{\left (e x +d \right )^{\frac {3}{2}}}d x\]
Input:
int((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(3/2),x)
Output:
int((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(3/2),x)
\[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="fricas")
Output:
integral(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*x + a*e)*(g*x + f)^n/sqrt(e*x + d), x)
Timed out. \[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)**n*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)** (3/2),x)
Output:
Timed out
\[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="maxima")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^n/(e*x + d)^(3/2), x)
\[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="giac")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^n/(e*x + d)^(3/2), x)
Timed out. \[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:
int(((f + g*x)^n*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x)^ (3/2),x)
Output:
int(((f + g*x)^n*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x)^ (3/2), x)
\[ \int \frac {(f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\text {too large to display} \] Input:
int((g*x+f)^n*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x)
Output:
(2*(4*(f + g*x)**n*sqrt(a*e + c*d*x)*a**2*e**2*f*g*n**2 + 12*(f + g*x)**n* sqrt(a*e + c*d*x)*a**2*e**2*f*g*n + 3*(f + g*x)**n*sqrt(a*e + c*d*x)*a**2* e**2*f*g + 4*(f + g*x)**n*sqrt(a*e + c*d*x)*a**2*e**2*g**2*n**2*x + 12*(f + g*x)**n*sqrt(a*e + c*d*x)*a**2*e**2*g**2*n*x - 4*(f + g*x)**n*sqrt(a*e + c*d*x)*a*c*d*e*f**2*n + 4*(f + g*x)**n*sqrt(a*e + c*d*x)*a*c*d*e*f*g*n**2 *x + 2*(f + g*x)**n*sqrt(a*e + c*d*x)*a*c*d*e*f*g*n*x + 6*(f + g*x)**n*sqr t(a*e + c*d*x)*a*c*d*e*f*g*x + 4*(f + g*x)**n*sqrt(a*e + c*d*x)*a*c*d*e*g* *2*n**2*x**2 + 6*(f + g*x)**n*sqrt(a*e + c*d*x)*a*c*d*e*g**2*n*x**2 + 2*(f + g*x)**n*sqrt(a*e + c*d*x)*c**2*d**2*f**2*n*x + 2*(f + g*x)**n*sqrt(a*e + c*d*x)*c**2*d**2*f*g*n*x**2 + 3*(f + g*x)**n*sqrt(a*e + c*d*x)*c**2*d**2 *f*g*x**2 + 24*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(8*a**2*e**2*f*g*n** 3 + 32*a**2*e**2*f*g*n**2 + 30*a**2*e**2*f*g*n + 8*a**2*e**2*g**2*n**3*x + 32*a**2*e**2*g**2*n**2*x + 30*a**2*e**2*g**2*n*x + 4*a*c*d*e*f**2*n**2 + 16*a*c*d*e*f**2*n + 15*a*c*d*e*f**2 + 8*a*c*d*e*f*g*n**3*x + 36*a*c*d*e*f* g*n**2*x + 46*a*c*d*e*f*g*n*x + 15*a*c*d*e*f*g*x + 8*a*c*d*e*g**2*n**3*x** 2 + 32*a*c*d*e*g**2*n**2*x**2 + 30*a*c*d*e*g**2*n*x**2 + 4*c**2*d**2*f**2* n**2*x + 16*c**2*d**2*f**2*n*x + 15*c**2*d**2*f**2*x + 4*c**2*d**2*f*g*n** 2*x**2 + 16*c**2*d**2*f*g*n*x**2 + 15*c**2*d**2*f*g*x**2),x)*a**4*e**4*g** 4*n**4 + 96*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(8*a**2*e**2*f*g*n**3 + 32*a**2*e**2*f*g*n**2 + 30*a**2*e**2*f*g*n + 8*a**2*e**2*g**2*n**3*x +...