Integrand size = 19, antiderivative size = 159 \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\frac {4}{(b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}-\frac {24 b (c+d x)^{3/4}}{5 (b c-a d)^2 (a+b x)^{5/4}}+\frac {48 \sqrt {b} d \sqrt [4]{\frac {d (a+b x)}{b (c+d x)}} \sqrt [4]{c+d x} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {-b c+a d}}{\sqrt {b} \sqrt {c+d x}}\right )\right |2\right )}{5 (-b c+a d)^{5/2} \sqrt [4]{a+b x}} \] Output:
4/(-a*d+b*c)/(b*x+a)^(5/4)/(d*x+c)^(1/4)-24/5*b*(d*x+c)^(3/4)/(-a*d+b*c)^2 /(b*x+a)^(5/4)+48/5*b^(1/2)*d*(d*(b*x+a)/b/(d*x+c))^(1/4)*(d*x+c)^(1/4)*El lipticE(sin(1/2*arctan((a*d-b*c)^(1/2)/b^(1/2)/(d*x+c)^(1/2))),2^(1/2))/(a *d-b*c)^(5/2)/(b*x+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=-\frac {4 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {5}{4},-\frac {1}{4},\frac {d (a+b x)}{-b c+a d}\right )}{5 b (a+b x)^{5/4} (c+d x)^{5/4}} \] Input:
Integrate[1/((a + b*x)^(9/4)*(c + d*x)^(5/4)),x]
Output:
(-4*((b*(c + d*x))/(b*c - a*d))^(5/4)*Hypergeometric2F1[-5/4, 5/4, -1/4, ( d*(a + b*x))/(-(b*c) + a*d)])/(5*b*(a + b*x)^(5/4)*(c + d*x)^(5/4))
Time = 0.40 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {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.
\(\displaystyle \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {6 d \int \frac {1}{(a+b x)^{5/4} (c+d x)^{5/4}}dx}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{5/4}}dx}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {2 b \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}}dx}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {8 \int \frac {\sqrt {a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [4]{a+b x}}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 839 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {8 \left (\frac {(a+b x)^{3/4}}{2 \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}-\frac {1}{2} \left (c-\frac {a d}{b}\right ) \int \frac {\sqrt {a+b x}}{\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{5/4}}d\sqrt [4]{a+b x}\right )}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {8 \left (\frac {(a+b x)^{3/4}}{2 \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}-\frac {b \sqrt [4]{a+b x} \left (c-\frac {a d}{b}\right ) \sqrt [4]{\frac {b c-a d}{d (a+b x)}+1} \int \frac {1}{(a+b x)^{3/4} \left (\frac {b c-a d}{d (a+b x)}+1\right )^{5/4}}d\sqrt [4]{a+b x}}{2 d \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {8 \left (\frac {b \sqrt [4]{a+b x} \left (c-\frac {a d}{b}\right ) \sqrt [4]{\frac {b c-a d}{d (a+b x)}+1} \int \frac {1}{\sqrt [4]{a+b x} \left (\frac {(b c-a d) (a+b x)}{d}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{a+b x}}}{2 d \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\frac {(a+b x)^{3/4}}{2 \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {8 \left (\frac {b \sqrt [4]{a+b x} \left (c-\frac {a d}{b}\right ) \sqrt [4]{\frac {b c-a d}{d (a+b x)}+1} \int \frac {1}{\left (\frac {\sqrt {a+b x} (b c-a d)}{d}+1\right )^{5/4}}d\sqrt {a+b x}}{4 d \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\frac {(a+b x)^{3/4}}{2 \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle -\frac {6 d \left (-\frac {2 d \left (\frac {4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {8 \left (\frac {b \sqrt [4]{a+b x} \left (c-\frac {a d}{b}\right ) \sqrt [4]{\frac {b c-a d}{d (a+b x)}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b c-a d} \sqrt {a+b x}}{\sqrt {d}}\right )\right |2\right )}{2 \sqrt {d} \sqrt {b c-a d} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\frac {(a+b x)^{3/4}}{2 \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4}{5 (a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)}\) |
Input:
Int[1/((a + b*x)^(9/4)*(c + d*x)^(5/4)),x]
Output:
-4/(5*(b*c - a*d)*(a + b*x)^(5/4)*(c + d*x)^(1/4)) - (6*d*(-4/((b*c - a*d) *(a + b*x)^(1/4)*(c + d*x)^(1/4)) - (2*d*((4*(a + b*x)^(3/4))/((b*c - a*d) *(c + d*x)^(1/4)) - (8*((a + b*x)^(3/4)/(2*(c - (a*d)/b + (d*(a + b*x))/b) ^(1/4)) + (b*(c - (a*d)/b)*(a + b*x)^(1/4)*(1 + (b*c - a*d)/(d*(a + b*x))) ^(1/4)*EllipticE[ArcTan[(Sqrt[b*c - a*d]*Sqrt[a + b*x])/Sqrt[d]]/2, 2])/(2 *Sqrt[d]*Sqrt[b*c - a*d]*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))))/(b*c - a *d)))/(b*c - a*d)))/(5*(b*c - a*d))
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]
\[\int \frac {1}{\left (b x +a \right )^{\frac {9}{4}} \left (x d +c \right )^{\frac {5}{4}}}d x\]
Input:
int(1/(b*x+a)^(9/4)/(d*x+c)^(5/4),x)
Output:
int(1/(b*x+a)^(9/4)/(d*x+c)^(5/4),x)
\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/(b*x+a)^(9/4)/(d*x+c)^(5/4),x, algorithm="fricas")
Output:
integral((b*x + a)^(3/4)*(d*x + c)^(3/4)/(b^3*d^2*x^5 + a^3*c^2 + (2*b^3*c *d + 3*a*b^2*d^2)*x^4 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^3 + (3*a*b ^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*x^2 + (3*a^2*b*c^2 + 2*a^3*c*d)*x), x)
\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {9}{4}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \] Input:
integrate(1/(b*x+a)**(9/4)/(d*x+c)**(5/4),x)
Output:
Integral(1/((a + b*x)**(9/4)*(c + d*x)**(5/4)), x)
\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/(b*x+a)^(9/4)/(d*x+c)^(5/4),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(9/4)*(d*x + c)^(5/4)), x)
\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/(b*x+a)^(9/4)/(d*x+c)^(5/4),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(9/4)*(d*x + c)^(5/4)), x)
Timed out. \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{9/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \] Input:
int(1/((a + b*x)^(9/4)*(c + d*x)^(5/4)),x)
Output:
int(1/((a + b*x)^(9/4)*(c + d*x)^(5/4)), x)
Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\frac {4 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \left (4 b d x +5 a d -b c \right )}{5 \sqrt {d x +c}\, \sqrt {b x +a}\, \left (a^{2} b \,d^{2} x -2 a \,b^{2} c d x +b^{3} c^{2} x +a^{3} d^{2}-2 a^{2} b c d +a \,b^{2} c^{2}\right )} \] Input:
int(1/(b*x+a)^(9/4)/(d*x+c)^(5/4),x)
Output:
(4*(c + d*x)**(1/4)*(a + b*x)**(1/4)*(5*a*d - b*c + 4*b*d*x))/(5*sqrt(c + d*x)*sqrt(a + b*x)*(a**3*d**2 - 2*a**2*b*c*d + a**2*b*d**2*x + a*b**2*c**2 - 2*a*b**2*c*d*x + b**3*c**2*x))