Integrand size = 15, antiderivative size = 126 \[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\frac {80 a^2 \sqrt {x} \sqrt [4]{a+b x}}{77 b^3}-\frac {40 a x^{3/2} \sqrt [4]{a+b x}}{77 b^2}+\frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {160 a^{7/2} \left (1+\frac {b x}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),2\right )}{77 b^{7/2} (a+b x)^{3/4}} \] Output:
80/77*a^2*x^(1/2)*(b*x+a)^(1/4)/b^3-40/77*a*x^(3/2)*(b*x+a)^(1/4)/b^2+4/11 *x^(5/2)*(b*x+a)^(1/4)/b-160/77*a^(7/2)*(1+b*x/a)^(3/4)*InverseJacobiAM(1/ 2*arctan(b^(1/2)*x^(1/2)/a^(1/2)),2^(1/2))/b^(7/2)/(b*x+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.37 \[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\frac {2 x^{7/2} \left (1+\frac {b x}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {9}{2},-\frac {b x}{a}\right )}{7 (a+b x)^{3/4}} \] Input:
Integrate[x^(5/2)/(a + b*x)^(3/4),x]
Output:
(2*x^(7/2)*(1 + (b*x)/a)^(3/4)*Hypergeometric2F1[3/4, 7/2, 9/2, -((b*x)/a) ])/(7*(a + b*x)^(3/4))
Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {60, 60, 60, 73, 765, 762}
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.
\(\displaystyle \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {10 a \int \frac {x^{3/2}}{(a+b x)^{3/4}}dx}{11 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {10 a \left (\frac {4 x^{3/2} \sqrt [4]{a+b x}}{7 b}-\frac {6 a \int \frac {\sqrt {x}}{(a+b x)^{3/4}}dx}{7 b}\right )}{11 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {10 a \left (\frac {4 x^{3/2} \sqrt [4]{a+b x}}{7 b}-\frac {6 a \left (\frac {4 \sqrt {x} \sqrt [4]{a+b x}}{3 b}-\frac {2 a \int \frac {1}{\sqrt {x} (a+b x)^{3/4}}dx}{3 b}\right )}{7 b}\right )}{11 b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {10 a \left (\frac {4 x^{3/2} \sqrt [4]{a+b x}}{7 b}-\frac {6 a \left (\frac {4 \sqrt {x} \sqrt [4]{a+b x}}{3 b}-\frac {8 a \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}}{3 b^2}\right )}{7 b}\right )}{11 b}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {10 a \left (\frac {4 x^{3/2} \sqrt [4]{a+b x}}{7 b}-\frac {6 a \left (\frac {4 \sqrt {x} \sqrt [4]{a+b x}}{3 b}-\frac {8 a \sqrt {1-\frac {a+b x}{a}} \int \frac {1}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{3 b^2 \sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{7 b}\right )}{11 b}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {4 x^{5/2} \sqrt [4]{a+b x}}{11 b}-\frac {10 a \left (\frac {4 x^{3/2} \sqrt [4]{a+b x}}{7 b}-\frac {6 a \left (\frac {4 \sqrt {x} \sqrt [4]{a+b x}}{3 b}-\frac {8 a^{5/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{3 b^2 \sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{7 b}\right )}{11 b}\) |
Input:
Int[x^(5/2)/(a + b*x)^(3/4),x]
Output:
(4*x^(5/2)*(a + b*x)^(1/4))/(11*b) - (10*a*((4*x^(3/2)*(a + b*x)^(1/4))/(7 *b) - (6*a*((4*Sqrt[x]*(a + b*x)^(1/4))/(3*b) - (8*a^(5/4)*Sqrt[1 - (a + b *x)/a]*EllipticF[ArcSin[(a + b*x)^(1/4)/a^(1/4)], -1])/(3*b^2*Sqrt[-(a/b) + (a + b*x)/b])))/(7*b)))/(11*b)
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[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
\[\int \frac {x^{\frac {5}{2}}}{\left (b x +a \right )^{\frac {3}{4}}}d x\]
Input:
int(x^(5/2)/(b*x+a)^(3/4),x)
Output:
int(x^(5/2)/(b*x+a)^(3/4),x)
\[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (b x + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^(5/2)/(b*x+a)^(3/4),x, algorithm="fricas")
Output:
integral(x^(5/2)/(b*x + a)^(3/4), x)
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.23 \[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\frac {2 x^{\frac {7}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{7 a^{\frac {3}{4}}} \] Input:
integrate(x**(5/2)/(b*x+a)**(3/4),x)
Output:
2*x**(7/2)*hyper((3/4, 7/2), (9/2,), b*x*exp_polar(I*pi)/a)/(7*a**(3/4))
\[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (b x + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^(5/2)/(b*x+a)^(3/4),x, algorithm="maxima")
Output:
integrate(x^(5/2)/(b*x + a)^(3/4), x)
\[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (b x + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^(5/2)/(b*x+a)^(3/4),x, algorithm="giac")
Output:
integrate(x^(5/2)/(b*x + a)^(3/4), x)
Timed out. \[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\int \frac {x^{5/2}}{{\left (a+b\,x\right )}^{3/4}} \,d x \] Input:
int(x^(5/2)/(a + b*x)^(3/4),x)
Output:
int(x^(5/2)/(a + b*x)^(3/4), x)
\[ \int \frac {x^{5/2}}{(a+b x)^{3/4}} \, dx=\int \frac {\sqrt {x}\, x^{2}}{\left (b x +a \right )^{\frac {3}{4}}}d x \] Input:
int(x^(5/2)/(b*x+a)^(3/4),x)
Output:
int((sqrt(x)*x**2)/(a + b*x)**(3/4),x)