Integrand size = 19, antiderivative size = 262 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=-\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}+\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}}-\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}} \]
-2/(-a*d+b*c)/(d*x+c)^(5/4)/(b*x+a)^(1/2)-14/5*d*(b*x+a)^(1/2)/(-a*d+b*c)^ 2/(d*x+c)^(5/4)-42/5*b*d*(b*x+a)^(1/2)/(-a*d+b*c)^3/(d*x+c)^(1/4)+42/5*b^( 5/4)*EllipticE(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a)/(-a*d +b*c))^(1/2)/(-a*d+b*c)^(9/4)/(b*x+a)^(1/2)-42/5*b^(5/4)*EllipticF(b^(1/4) *(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a)/(-a*d+b*c))^(1/2)/(-a*d+b*c )^(9/4)/(b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.27 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {9}{4},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} (c+d x)^{9/4}} \]
(-2*((b*(c + d*x))/(b*c - a*d))^(9/4)*Hypergeometric2F1[-1/2, 9/4, 1/2, (d *(a + b*x))/(-(b*c) + a*d)])/(b*Sqrt[a + b*x]*(c + d*x)^(9/4))
Time = 0.50 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {61, 61, 61, 73, 836, 765, 762, 1390, 1388, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}}dx}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}}dx}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {b \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}}dx}{b c-a d}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {7 d \left (\frac {3 b \left (\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac {4 b \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d (b c-a d)}\right )}{5 (b c-a d)}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)}\) |
-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(5/4)) - (7*d*((4*Sqrt[a + b*x])/( 5*(b*c - a*d)*(c + d*x)^(5/4)) + (3*b*((4*Sqrt[a + b*x])/((b*c - a*d)*(c + d*x)^(1/4)) - (4*b*(((b*c - a*d)^(3/4)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d) ]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^( 3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) - ((b*c - a*d)^(3/4)*Sqrt[1 - (b *(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])))/(d*(b*c - a*d))))/(5*(b*c - a*d))))/(2*(b*c - a*d))
3.17.76.3.1 Defintions of rubi rules used
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 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])
\[\int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {9}{4}}}d x\]
\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
integral(sqrt(b*x + a)*(d*x + c)^(3/4)/(b^2*d^3*x^5 + a^2*c^3 + (3*b^2*c*d ^2 + 2*a*b*d^3)*x^4 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^3 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^2 + (2*a*b*c^3 + 3*a^2*c^2*d)*x), x)
\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {9}{4}}}\, dx \]
\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{9/4}} \,d x \]
\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\text {too large to display} \]
int(1/((c + d*x)**(1/4)*sqrt(a + b*x)*(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b*c**2*x + 2*b*c*d*x**2 + b*d**2*x**3)),x)
( - 4*(c + d*x)**(1/4)*sqrt(a + b*x) - 15*sqrt(c + d*x)*int(((c + d*x)**(3 /4)*sqrt(a + b*x)*x)/(3*a**3*c**3*d + 9*a**3*c**2*d**2*x + 9*a**3*c*d**3*x **2 + 3*a**3*d**4*x**3 + 2*a**2*b*c**4 + 12*a**2*b*c**3*d*x + 24*a**2*b*c* *2*d**2*x**2 + 20*a**2*b*c*d**3*x**3 + 6*a**2*b*d**4*x**4 + 4*a*b**2*c**4* x + 15*a*b**2*c**3*d*x**2 + 21*a*b**2*c**2*d**2*x**3 + 13*a*b**2*c*d**3*x* *4 + 3*a*b**2*d**4*x**5 + 2*b**3*c**4*x**2 + 6*b**3*c**3*d*x**3 + 6*b**3*c **2*d**2*x**4 + 2*b**3*c*d**3*x**5),x)*a**2*b*c*d**2 - 15*sqrt(c + d*x)*in t(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**3*c**3*d + 9*a**3*c**2*d**2*x + 9*a**3*c*d**3*x**2 + 3*a**3*d**4*x**3 + 2*a**2*b*c**4 + 12*a**2*b*c**3*d* x + 24*a**2*b*c**2*d**2*x**2 + 20*a**2*b*c*d**3*x**3 + 6*a**2*b*d**4*x**4 + 4*a*b**2*c**4*x + 15*a*b**2*c**3*d*x**2 + 21*a*b**2*c**2*d**2*x**3 + 13* a*b**2*c*d**3*x**4 + 3*a*b**2*d**4*x**5 + 2*b**3*c**4*x**2 + 6*b**3*c**3*d *x**3 + 6*b**3*c**2*d**2*x**4 + 2*b**3*c*d**3*x**5),x)*a**2*b*d**3*x - 10* sqrt(c + d*x)*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**3*c**3*d + 9*a* *3*c**2*d**2*x + 9*a**3*c*d**3*x**2 + 3*a**3*d**4*x**3 + 2*a**2*b*c**4 + 1 2*a**2*b*c**3*d*x + 24*a**2*b*c**2*d**2*x**2 + 20*a**2*b*c*d**3*x**3 + 6*a **2*b*d**4*x**4 + 4*a*b**2*c**4*x + 15*a*b**2*c**3*d*x**2 + 21*a*b**2*c**2 *d**2*x**3 + 13*a*b**2*c*d**3*x**4 + 3*a*b**2*d**4*x**5 + 2*b**3*c**4*x**2 + 6*b**3*c**3*d*x**3 + 6*b**3*c**2*d**2*x**4 + 2*b**3*c*d**3*x**5),x)*a*b **2*c**2*d - 25*sqrt(c + d*x)*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3...