Integrand size = 15, antiderivative size = 77 \[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=-\frac {2}{a \sqrt {x} \sqrt [4]{a+b x}}-\frac {6 \sqrt {b} \sqrt [4]{\frac {a+b x}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x}} \] Output:
-2/a/x^(1/2)/(b*x+a)^(1/4)-6*b^(1/2)*((b*x+a)/a)^(1/4)*EllipticE(sin(1/2*a rctan(b^(1/2)*x^(1/2)/a^(1/2))),2^(1/2))/a^(3/2)/(b*x+a)^(1/4)
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 = 0.62 \[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=-\frac {2 \sqrt [4]{1+\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{4},\frac {1}{2},-\frac {b x}{a}\right )}{a \sqrt {x} \sqrt [4]{a+b x}} \] Input:
Integrate[1/(x^(3/2)*(a + b*x)^(5/4)),x]
Output:
(-2*(1 + (b*x)/a)^(1/4)*Hypergeometric2F1[-1/2, 5/4, 1/2, -((b*x)/a)])/(a* Sqrt[x]*(a + b*x)^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(77)=154\).
Time = 0.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.23, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {61, 61, 73, 836, 27, 765, 762, 1390, 1389, 327}
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.
\(\displaystyle \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 \int \frac {1}{x^{3/2} \sqrt [4]{a+b x}}dx}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 \left (\frac {b \int \frac {1}{\sqrt {x} \sqrt [4]{a+b x}}dx}{2 a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 \left (\frac {2 \int \frac {\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\sqrt {a} \int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {a} \sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\sqrt {a} \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\sqrt {a} \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\frac {\sqrt {a} \sqrt {1-\frac {a+b x}{a}} \int \frac {1}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 1390 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\sqrt {1-\frac {a+b x}{a}} \int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\sqrt {a} \sqrt {1-\frac {a+b x}{a}} \int \frac {\sqrt {\frac {\sqrt {a+b x}}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {a+b x}}{\sqrt {a}}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{a}-\frac {2 (a+b x)^{3/4}}{a \sqrt {x}}\right )}{a}+\frac {4}{a \sqrt {x} \sqrt [4]{a+b x}}\) |
Input:
Int[1/(x^(3/2)*(a + b*x)^(5/4)),x]
Output:
4/(a*Sqrt[x]*(a + b*x)^(1/4)) + (3*((-2*(a + b*x)^(3/4))/(a*Sqrt[x]) + (2* ((a^(3/4)*Sqrt[1 - (a + b*x)/a]*EllipticE[ArcSin[(a + b*x)^(1/4)/a^(1/4)], -1])/Sqrt[-(a/b) + (a + b*x)/b] - (a^(3/4)*Sqrt[1 - (a + b*x)/a]*Elliptic F[ArcSin[(a + b*x)^(1/4)/a^(1/4)], -1])/Sqrt[-(a/b) + (a + b*x)/b]))/a))/a
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
\[\int \frac {1}{x^{\frac {3}{2}} \left (b x +a \right )^{\frac {5}{4}}}d x\]
Input:
int(1/x^(3/2)/(b*x+a)^(5/4),x)
Output:
int(1/x^(3/2)/(b*x+a)^(5/4),x)
\[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} x^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/x^(3/2)/(b*x+a)^(5/4),x, algorithm="fricas")
Output:
integral((b*x + a)^(3/4)*sqrt(x)/(b^2*x^4 + 2*a*b*x^3 + a^2*x^2), x)
Result contains complex when optimal does not.
Time = 1.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=- \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {5}{4}} \sqrt {x}} \] Input:
integrate(1/x**(3/2)/(b*x+a)**(5/4),x)
Output:
-2*hyper((-1/2, 5/4), (1/2,), b*x*exp_polar(I*pi)/a)/(a**(5/4)*sqrt(x))
\[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} x^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/x^(3/2)/(b*x+a)^(5/4),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(5/4)*x^(3/2)), x)
\[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} x^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/x^(3/2)/(b*x+a)^(5/4),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(5/4)*x^(3/2)), x)
Timed out. \[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=\int \frac {1}{x^{3/2}\,{\left (a+b\,x\right )}^{5/4}} \,d x \] Input:
int(1/(x^(3/2)*(a + b*x)^(5/4)),x)
Output:
int(1/(x^(3/2)*(a + b*x)^(5/4)), x)
\[ \int \frac {1}{x^{3/2} (a+b x)^{5/4}} \, dx=\int \frac {\sqrt {x}\, \left (b x +a \right )^{\frac {1}{4}}}{\sqrt {b x +a}\, a \,x^{2}+\sqrt {b x +a}\, b \,x^{3}}d x \] Input:
int(1/x^(3/2)/(b*x+a)^(5/4),x)
Output:
int((sqrt(x)*(a + b*x)**(1/4))/(sqrt(a + b*x)*a*x**2 + sqrt(a + b*x)*b*x** 3),x)