Integrand size = 19, antiderivative size = 118 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\frac {4 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/4}}+\frac {4 b^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d (b c-a d)^{3/4} \sqrt {a+b x}} \] Output:
4/3*(b*x+a)^(1/2)/(-a*d+b*c)/(d*x+c)^(3/4)+4/3*b^(3/4)*(-d*(b*x+a)/(-a*d+b *c))^(1/2)*EllipticF(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)/d/(-a*d+b*c )^(3/4)/(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 = 0.60 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{7/4}} \] Input:
Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(7/4)),x]
Output:
(2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(7/4)*Hypergeometric2F1[1/2, 7/4, 3/2, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(c + d*x)^(7/4))
Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {b \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}}dx}{3 (b c-a d)}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 b \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{3 d (b c-a d)}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {4 b \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{3 d (b c-a d) \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {4 b^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d (b c-a d)^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}+\frac {4 \sqrt {a+b x}}{3 (c+d x)^{3/4} (b c-a d)}\) |
Input:
Int[1/(Sqrt[a + b*x]*(c + d*x)^(7/4)),x]
Output:
(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])
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]
\[\int \frac {1}{\sqrt {b x +a}\, \left (x d +c \right )^{\frac {7}{4}}}d x\]
Input:
int(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x)
Output:
int(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x)
\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x, algorithm="fricas")
Output:
integral(sqrt(b*x + a)*(d*x + c)^(1/4)/(b*d^2*x^3 + a*c^2 + (2*b*c*d + a*d ^2)*x^2 + (b*c^2 + 2*a*c*d)*x), x)
\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {7}{4}}}\, dx \] Input:
integrate(1/(b*x+a)**(1/2)/(d*x+c)**(7/4),x)
Output:
Integral(1/(sqrt(a + b*x)*(c + d*x)**(7/4)), x)
\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/4)), x)
\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/4)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{7/4}} \,d x \] Input:
int(1/((a + b*x)^(1/2)*(c + d*x)^(7/4)),x)
Output:
int(1/((a + b*x)^(1/2)*(c + d*x)^(7/4)), x)
\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx=\frac {-4 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}+\sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{3} x^{3}-2 b^{2} c \,d^{2} x^{3}+a^{2} d^{3} x^{2}-4 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2} x -3 a b \,c^{2} d x -2 b^{2} c^{3} x +a^{2} c^{2} d -2 a b \,c^{3}}d x \right ) a b c \,d^{2}+\sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{3} x^{3}-2 b^{2} c \,d^{2} x^{3}+a^{2} d^{3} x^{2}-4 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2} x -3 a b \,c^{2} d x -2 b^{2} c^{3} x +a^{2} c^{2} d -2 a b \,c^{3}}d x \right ) a b \,d^{3} x -2 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{3} x^{3}-2 b^{2} c \,d^{2} x^{3}+a^{2} d^{3} x^{2}-4 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2} x -3 a b \,c^{2} d x -2 b^{2} c^{3} x +a^{2} c^{2} d -2 a b \,c^{3}}d x \right ) b^{2} c^{2} d -2 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{3} x^{3}-2 b^{2} c \,d^{2} x^{3}+a^{2} d^{3} x^{2}-4 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2} x -3 a b \,c^{2} d x -2 b^{2} c^{3} x +a^{2} c^{2} d -2 a b \,c^{3}}d x \right ) b^{2} c \,d^{2} x}{\sqrt {d x +c}\, \left (a \,d^{2} x -2 b c d x +a c d -2 b \,c^{2}\right )} \] Input:
int(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x)
Output:
( - 4*(c + d*x)**(3/4)*sqrt(a + b*x) + sqrt(c + d*x)*int(((c + d*x)**(1/4) *sqrt(a + b*x)*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - 2*a*b* c**3 - 3*a*b*c**2*d*x + a*b*d**3*x**3 - 2*b**2*c**3*x - 4*b**2*c**2*d*x**2 - 2*b**2*c*d**2*x**3),x)*a*b*c*d**2 + sqrt(c + d*x)*int(((c + d*x)**(1/4) *sqrt(a + b*x)*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - 2*a*b* c**3 - 3*a*b*c**2*d*x + a*b*d**3*x**3 - 2*b**2*c**3*x - 4*b**2*c**2*d*x**2 - 2*b**2*c*d**2*x**3),x)*a*b*d**3*x - 2*sqrt(c + d*x)*int(((c + d*x)**(1/ 4)*sqrt(a + b*x)*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - 2*a* b*c**3 - 3*a*b*c**2*d*x + a*b*d**3*x**3 - 2*b**2*c**3*x - 4*b**2*c**2*d*x* *2 - 2*b**2*c*d**2*x**3),x)*b**2*c**2*d - 2*sqrt(c + d*x)*int(((c + d*x)** (1/4)*sqrt(a + b*x)*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - 2 *a*b*c**3 - 3*a*b*c**2*d*x + a*b*d**3*x**3 - 2*b**2*c**3*x - 4*b**2*c**2*d *x**2 - 2*b**2*c*d**2*x**3),x)*b**2*c*d**2*x)/(sqrt(c + d*x)*(a*c*d + a*d* *2*x - 2*b*c**2 - 2*b*c*d*x))