Integrand size = 19, antiderivative size = 178 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {5 d^2 \sqrt {a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac {5 b^{3/4} d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{(b c-a d)^{11/4} \sqrt {a+b x}} \]
-2/3/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/4)+3*d/(-a*d+b*c)^2/(d*x+c)^(3/4) /(b*x+a)^(1/2)+5*d^2*(b*x+a)^(1/2)/(-a*d+b*c)^3/(d*x+c)^(3/4)+5*b^(3/4)*d* 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)^(11/4)/(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 = 73, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{7/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {7}{4},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} (c+d x)^{7/4}} \]
(-2*((b*(c + d*x))/(b*c - a*d))^(7/4)*Hypergeometric2F1[-3/2, 7/4, -1/2, ( d*(a + b*x))/(-(b*c) + a*d)])/(3*b*(a + b*x)^(3/2)*(c + d*x)^(7/4))
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {61, 61, 61, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {3 d \int \frac {1}{(a+b x)^{3/2} (c+d x)^{7/4}}dx}{2 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {3 d \left (-\frac {5 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}}dx}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {3 d \left (-\frac {5 d \left (\frac {b \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}}dx}{3 (b c-a d)}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {3 d \left (-\frac {5 d \left (\frac {4 b \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{3 d (b c-a d)}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle -\frac {3 d \left (-\frac {5 d \left (\frac {4 b \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{3 d (b c-a d) \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle -\frac {3 d \left (-\frac {5 d \left (\frac {4 b^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d (b c-a d)^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}\right )}{2 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}\) |
-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/4)) - (3*d*(-2/((b*c - a*d) *Sqrt[a + b*x]*(c + d*x)^(3/4)) - (5*d*((4*Sqrt[a + b*x])/(3*(b*c - a*d)*( c + d*x)^(3/4)) + (4*b^(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])/(3*d*(b*c - a*d )^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])))/(2*(b*c - a*d))))/(2*(b*c - a*d))
3.17.70.3.1 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[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]
\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {7}{4}}}d x\]
\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \]
integral(sqrt(b*x + a)*(d*x + c)^(1/4)/(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)
\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {7}{4}}}\, dx \]
\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \]
\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{7/4}} \,d x \]
\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\text {too large to display} \]
int(1/((c + d*x)**(3/4)*sqrt(a + b*x)*(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a *b*d*x**2 + b**2*c*x**2 + b**2*d*x**3)),x)
( - 4*(c + d*x)**(3/4)*sqrt(a + b*x) - 7*sqrt(c + d*x)*int(((c + d*x)**(1/ 4)*sqrt(a + b*x)*x)/(a**4*c**2*d + 2*a**4*c*d**2*x + a**4*d**3*x**2 + 6*a* *3*b*c**3 + 15*a**3*b*c**2*d*x + 12*a**3*b*c*d**2*x**2 + 3*a**3*b*d**3*x** 3 + 18*a**2*b**2*c**3*x + 39*a**2*b**2*c**2*d*x**2 + 24*a**2*b**2*c*d**2*x **3 + 3*a**2*b**2*d**3*x**4 + 18*a*b**3*c**3*x**2 + 37*a*b**3*c**2*d*x**3 + 20*a*b**3*c*d**2*x**4 + a*b**3*d**3*x**5 + 6*b**4*c**3*x**3 + 12*b**4*c* *2*d*x**4 + 6*b**4*c*d**2*x**5),x)*a**3*b*c*d**2 - 7*sqrt(c + d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**4*c**2*d + 2*a**4*c*d**2*x + a**4*d**3 *x**2 + 6*a**3*b*c**3 + 15*a**3*b*c**2*d*x + 12*a**3*b*c*d**2*x**2 + 3*a** 3*b*d**3*x**3 + 18*a**2*b**2*c**3*x + 39*a**2*b**2*c**2*d*x**2 + 24*a**2*b **2*c*d**2*x**3 + 3*a**2*b**2*d**3*x**4 + 18*a*b**3*c**3*x**2 + 37*a*b**3* c**2*d*x**3 + 20*a*b**3*c*d**2*x**4 + a*b**3*d**3*x**5 + 6*b**4*c**3*x**3 + 12*b**4*c**2*d*x**4 + 6*b**4*c*d**2*x**5),x)*a**3*b*d**3*x - 42*sqrt(c + d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**4*c**2*d + 2*a**4*c*d**2* x + a**4*d**3*x**2 + 6*a**3*b*c**3 + 15*a**3*b*c**2*d*x + 12*a**3*b*c*d**2 *x**2 + 3*a**3*b*d**3*x**3 + 18*a**2*b**2*c**3*x + 39*a**2*b**2*c**2*d*x** 2 + 24*a**2*b**2*c*d**2*x**3 + 3*a**2*b**2*d**3*x**4 + 18*a*b**3*c**3*x**2 + 37*a*b**3*c**2*d*x**3 + 20*a*b**3*c*d**2*x**4 + a*b**3*d**3*x**5 + 6*b* *4*c**3*x**3 + 12*b**4*c**2*d*x**4 + 6*b**4*c*d**2*x**5),x)*a**2*b**2*c**2 *d - 56*sqrt(c + d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**4*c**2...