Integrand size = 17, antiderivative size = 188 \[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\frac {448 b^3}{39 a^4 \sqrt [4]{a x+b x^2}}+\frac {4}{a x^3 \sqrt [4]{a x+b x^2}}-\frac {56 \left (a x+b x^2\right )^{3/4}}{13 a^2 x^4}+\frac {560 b \left (a x+b x^2\right )^{3/4}}{117 a^3 x^3}-\frac {224 b^2 \left (a x+b x^2\right )^{3/4}}{39 a^4 x^2}-\frac {448 b^3 \sqrt [4]{\frac {b x}{a+b x}} \sqrt {a+b x} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right )\right |2\right )}{39 a^{9/2} \sqrt [4]{a x+b x^2}} \] Output:
448/39*b^3/a^4/(b*x^2+a*x)^(1/4)+4/a/x^3/(b*x^2+a*x)^(1/4)-56/13*(b*x^2+a* x)^(3/4)/a^2/x^4+560/117*b*(b*x^2+a*x)^(3/4)/a^3/x^3-224/39*b^2*(b*x^2+a*x )^(3/4)/a^4/x^2-448/39*b^3*(b*x/(b*x+a))^(1/4)*(b*x+a)^(1/2)*EllipticE(sin (1/2*arcsin(a^(1/2)/(b*x+a)^(1/2))),2^(1/2))/a^(9/2)/(b*x^2+a*x)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=-\frac {4 \sqrt [4]{1+\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {5}{4},-\frac {9}{4},-\frac {b x}{a}\right )}{13 a x^3 \sqrt [4]{x (a+b x)}} \] Input:
Integrate[1/(x^3*(a*x + b*x^2)^(5/4)),x]
Output:
(-4*(1 + (b*x)/a)^(1/4)*Hypergeometric2F1[-13/4, 5/4, -9/4, -((b*x)/a)])/( 13*a*x^3*(x*(a + b*x))^(1/4))
Time = 0.66 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1137, 61, 61, 61, 61, 61, 73, 839, 813, 858, 807, 212}
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.
\(\displaystyle \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx\) |
\(\Big \downarrow \) 1137 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \int \frac {1}{x^{17/4} (a+b x)^{5/4}}dx}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \int \frac {1}{x^{17/4} \sqrt [4]{a+b x}}dx}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \int \frac {1}{x^{13/4} \sqrt [4]{a+b x}}dx}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \int \frac {1}{x^{9/4} \sqrt [4]{a+b x}}dx}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \int \frac {1}{x^{5/4} \sqrt [4]{a+b x}}dx}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {2 b \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}}dx}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \int \frac {\sqrt {x}}{\sqrt [4]{a+b x}}d\sqrt [4]{x}}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 839 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {1}{2} a \int \frac {\sqrt {x}}{(a+b x)^{5/4}}d\sqrt [4]{x}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{5/4} x^{3/4}}d\sqrt [4]{x}}{2 b \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{x}}}{2 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{5/4}}d\sqrt {x}}{4 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {x^{5/4} (a+b x)^{5/4} \left (\frac {14 \left (-\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {\sqrt {a} \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |2\right )}{2 \sqrt {b} \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{13 a}-\frac {4 (a+b x)^{3/4}}{13 a x^{13/4}}\right )}{a}+\frac {4}{a x^{13/4} \sqrt [4]{a+b x}}\right )}{\left (a x+b x^2\right )^{5/4}}\) |
Input:
Int[1/(x^3*(a*x + b*x^2)^(5/4)),x]
Output:
(x^(5/4)*(a + b*x)^(5/4)*(4/(a*x^(13/4)*(a + b*x)^(1/4)) + (14*((-4*(a + b *x)^(3/4))/(13*a*x^(13/4)) - (10*b*((-4*(a + b*x)^(3/4))/(9*a*x^(9/4)) - ( 2*b*((-4*(a + b*x)^(3/4))/(5*a*x^(5/4)) - (2*b*((-4*(a + b*x)^(3/4))/(a*x^ (1/4)) + (8*b*(x^(3/4)/(2*(a + b*x)^(1/4)) + (Sqrt[a]*(1 + a/(b*x))^(1/4)* x^(1/4)*EllipticE[ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/2, 2])/(2*Sqrt[b]*(a + b*x)^(1/4))))/a))/(5*a)))/(3*a)))/(13*a)))/a))/(a*x + b*x^2)^(5/4)
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[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
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_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 )^(1/4)), x] - Simp[a/2 Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} , x] && PosQ[b/a]
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]
\[\int \frac {1}{x^{3} \left (b \,x^{2}+a x \right )^{\frac {5}{4}}}d x\]
Input:
int(1/x^3/(b*x^2+a*x)^(5/4),x)
Output:
int(1/x^3/(b*x^2+a*x)^(5/4),x)
\[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {5}{4}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(b*x^2+a*x)^(5/4),x, algorithm="fricas")
Output:
integral((b*x^2 + a*x)^(3/4)/(b^2*x^7 + 2*a*b*x^6 + a^2*x^5), x)
\[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\int \frac {1}{x^{3} \left (x \left (a + b x\right )\right )^{\frac {5}{4}}}\, dx \] Input:
integrate(1/x**3/(b*x**2+a*x)**(5/4),x)
Output:
Integral(1/(x**3*(x*(a + b*x))**(5/4)), x)
\[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {5}{4}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(b*x^2+a*x)^(5/4),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a*x)^(5/4)*x^3), x)
\[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {5}{4}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(b*x^2+a*x)^(5/4),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a*x)^(5/4)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\int \frac {1}{x^3\,{\left (b\,x^2+a\,x\right )}^{5/4}} \,d x \] Input:
int(1/(x^3*(a*x + b*x^2)^(5/4)),x)
Output:
int(1/(x^3*(a*x + b*x^2)^(5/4)), x)
Time = 0.42 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^3 \left (a x+b x^2\right )^{5/4}} \, dx=\frac {4 \left (b x +a \right )^{\frac {1}{4}} \left (-128 b^{3} x^{3}-32 a \,b^{2} x^{2}+12 a^{2} b x -7 a^{3}\right )}{77 x^{\frac {11}{4}} \sqrt {x}\, \sqrt {b x +a}\, a^{4}} \] Input:
int(1/x^3/(b*x^2+a*x)^(5/4),x)
Output:
(4*x**(1/4)*(a + b*x)**(1/4)*( - 7*a**3 + 12*a**2*b*x - 32*a*b**2*x**2 - 1 28*b**3*x**3))/(77*sqrt(x)*sqrt(a + b*x)*a**4*x**3)