Integrand size = 17, antiderivative size = 214 \[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=-\frac {65 a^5 \sqrt [4]{a x+b x^2}}{3696 b^4}+\frac {13 a^4 x \sqrt [4]{a x+b x^2}}{1848 b^3}-\frac {13 a^3 x^2 \sqrt [4]{a x+b x^2}}{2772 b^2}+\frac {5 a^2 x^3 \sqrt [4]{a x+b x^2}}{1386 b}+\frac {5}{99} a x^4 \sqrt [4]{a x+b x^2}+\frac {2}{11} x^3 \left (a x+b x^2\right )^{5/4}-\frac {65 a^{11/2} \left (\frac {b x}{a+b x}\right )^{3/4} \sqrt {a+b x} \sqrt [4]{a x+b x^2} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right ),2\right )}{3696 b^5 x} \] Output:
-65/3696*a^5*(b*x^2+a*x)^(1/4)/b^4+13/1848*a^4*x*(b*x^2+a*x)^(1/4)/b^3-13/ 2772*a^3*x^2*(b*x^2+a*x)^(1/4)/b^2+5/1386*a^2*x^3*(b*x^2+a*x)^(1/4)/b+5/99 *a*x^4*(b*x^2+a*x)^(1/4)+2/11*x^3*(b*x^2+a*x)^(5/4)-65/3696*a^(11/2)*(b*x/ (b*x+a))^(3/4)*(b*x+a)^(1/2)*(b*x^2+a*x)^(1/4)*InverseJacobiAM(1/2*arcsin( a^(1/2)/(b*x+a)^(1/2)),2^(1/2))/b^5/x
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.22 \[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\frac {4 a x^4 \sqrt [4]{x (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {17}{4},\frac {21}{4},-\frac {b x}{a}\right )}{17 \sqrt [4]{1+\frac {b x}{a}}} \] Input:
Integrate[x^2*(a*x + b*x^2)^(5/4),x]
Output:
(4*a*x^4*(x*(a + b*x))^(1/4)*Hypergeometric2F1[-5/4, 17/4, 21/4, -((b*x)/a )])/(17*(1 + (b*x)/a)^(1/4))
Time = 0.62 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1137, 60, 60, 60, 60, 60, 60, 73, 768, 858, 807, 229}
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.
| \(\displaystyle \int x^2 \left (a x+b x^2\right )^{5/4} \, dx\) |
| \(\Big \downarrow \) 1137 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \int x^{13/4} (a+b x)^{5/4}dx}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \int x^{13/4} \sqrt [4]{a+b x}dx+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \int \frac {x^{13/4}}{(a+b x)^{3/4}}dx+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \int \frac {x^{9/4}}{(a+b x)^{3/4}}dx}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \int \frac {x^{5/4}}{(a+b x)^{3/4}}dx}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \int \frac {\sqrt [4]{x}}{(a+b x)^{3/4}}dx}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {a \int \frac {1}{x^{3/4} (a+b x)^{3/4}}dx}{2 b}\right )}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {2 a \int \frac {1}{(a+b x)^{3/4}}d\sqrt [4]{x}}{b}\right )}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 768 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {2 a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{3/4} x^{3/4}}d\sqrt [4]{x}}{b (a+b x)^{3/4}}\right )}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{3/4}}d\frac {1}{\sqrt [4]{x}}}{b (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{3/4}}d\sqrt {x}}{b (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
| \(\Big \downarrow \) 229 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/4} \left (\frac {5}{22} a \left (\frac {1}{18} a \left (\frac {2 x^{13/4} \sqrt [4]{a+b x}}{7 b}-\frac {13 a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt {a} x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),2\right )}{\sqrt {b} (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )}{10 b}\right )}{14 b}\right )+\frac {2}{9} x^{17/4} \sqrt [4]{a+b x}\right )+\frac {2}{11} x^{17/4} (a+b x)^{5/4}\right )}{x^{5/4} (a+b x)^{5/4}}\) |
Input:
Int[x^2*(a*x + b*x^2)^(5/4),x]
Output:
((a*x + b*x^2)^(5/4)*((2*x^(17/4)*(a + b*x)^(5/4))/11 + (5*a*((2*x^(17/4)* (a + b*x)^(1/4))/9 + (a*((2*x^(13/4)*(a + b*x)^(1/4))/(7*b) - (13*a*((2*x^ (9/4)*(a + b*x)^(1/4))/(5*b) - (9*a*((2*x^(5/4)*(a + b*x)^(1/4))/(3*b) - ( 5*a*((2*x^(1/4)*(a + b*x)^(1/4))/b + (2*Sqrt[a]*(1 + a/(b*x))^(3/4)*x^(3/4 )*EllipticF[ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/2, 2])/(Sqrt[b]*(a + b*x)^(3 /4))))/(6*b)))/(10*b)))/(14*b)))/18))/22))/(x^(5/4)*(a + b*x)^(5/4))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[( e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(b + c*x)^ p, x], x] /; FreeQ[{b, c, e, m}, x]
\[\int x^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{4}}d x\]
Input:
int(x^2*(b*x^2+a*x)^(5/4),x)
Output:
int(x^2*(b*x^2+a*x)^(5/4),x)
\[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\int { {\left (b x^{2} + a x\right )}^{\frac {5}{4}} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^2+a*x)^(5/4),x, algorithm="fricas")
Output:
integral((b*x^4 + a*x^3)*(b*x^2 + a*x)^(1/4), x)
\[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\int x^{2} \left (x \left (a + b x\right )\right )^{\frac {5}{4}}\, dx \] Input:
integrate(x**2*(b*x**2+a*x)**(5/4),x)
Output:
Integral(x**2*(x*(a + b*x))**(5/4), x)
\[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\int { {\left (b x^{2} + a x\right )}^{\frac {5}{4}} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^2+a*x)^(5/4),x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(5/4)*x^2, x)
\[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\int { {\left (b x^{2} + a x\right )}^{\frac {5}{4}} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^2+a*x)^(5/4),x, algorithm="giac")
Output:
integrate((b*x^2 + a*x)^(5/4)*x^2, x)
Timed out. \[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\int x^2\,{\left (b\,x^2+a\,x\right )}^{5/4} \,d x \] Input:
int(x^2*(a*x + b*x^2)^(5/4),x)
Output:
int(x^2*(a*x + b*x^2)^(5/4), x)
\[ \int x^2 \left (a x+b x^2\right )^{5/4} \, dx=\frac {-780 x^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{5}+312 x^{\frac {5}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{4} b -208 x^{\frac {9}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{3} b^{2}+160 x^{\frac {13}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{2} b^{3}+10304 x^{\frac {17}{4}} \left (b x +a \right )^{\frac {1}{4}} a \,b^{4}+8064 x^{\frac {21}{4}} \left (b x +a \right )^{\frac {1}{4}} b^{5}+195 \left (\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{\frac {3}{4}} a +x^{\frac {7}{4}} b}d x \right ) a^{6}}{44352 b^{4}} \] Input:
int(x^2*(b*x^2+a*x)^(5/4),x)
Output:
( - 780*x**(1/4)*(a + b*x)**(1/4)*a**5 + 312*x**(1/4)*(a + b*x)**(1/4)*a** 4*b*x - 208*x**(1/4)*(a + b*x)**(1/4)*a**3*b**2*x**2 + 160*x**(1/4)*(a + b *x)**(1/4)*a**2*b**3*x**3 + 10304*x**(1/4)*(a + b*x)**(1/4)*a*b**4*x**4 + 8064*x**(1/4)*(a + b*x)**(1/4)*b**5*x**5 + 195*int((a + b*x)**(1/4)/(x**(3 /4)*a + x**(3/4)*b*x),x)*a**6)/(44352*b**4)