\(\int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx\) [461]

Optimal result

Integrand size = 19, antiderivative size = 111 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=-\frac {2 \sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+b x}}-\frac {2 \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt [4]{b} (b c-a d)^{3/4} \sqrt {a+b x}} \] Output:

-2*(d*x+c)^(1/4)/(-a*d+b*c)/(b*x+a)^(1/2)-2*(-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)/b^(1/4)/(-a*d+b*c)^(3/ 
4)/(b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} (c+d x)^{3/4}} \] Input:

Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(3/4)),x]
 

Output:

(-2*((b*(c + d*x))/(b*c - a*d))^(3/4)*Hypergeometric2F1[-1/2, 3/4, 1/2, (d 
*(a + b*x))/(-(b*c) + a*d)])/(b*Sqrt[a + b*x]*(c + d*x)^(3/4))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 127, 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.

\(\displaystyle \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}}dx}{2 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {2 \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{b c-a d}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 765

\(\displaystyle -\frac {2 \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 762

\(\displaystyle -\frac {2 \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt [4]{b} (b c-a d)^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\)

Input:

Int[1/((a + b*x)^(3/2)*(c + d*x)^(3/4)),x]
 

Output:

(-2*(c + d*x)^(1/4))/((b*c - a*d)*Sqrt[a + b*x]) - (2*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^(1/4)*(b*c - a*d)^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])
 

Defintions of rubi rules used

rule 61

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]
 

rule 73

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]
 

rule 762

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]
 

rule 765

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]
 

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{4}}}d x\]

Input:

int(1/(b*x+a)^(3/2)/(d*x+c)^(3/4),x)
 

Output:

int(1/(b*x+a)^(3/2)/(d*x+c)^(3/4),x)
 

Fricas [F]

\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(3/4),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x + a)*(d*x + c)^(1/4)/(b^2*d*x^3 + a^2*c + (b^2*c + 2*a*b 
*d)*x^2 + (2*a*b*c + a^2*d)*x), x)
 

Sympy [F]

\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(1/(b*x+a)**(3/2)/(d*x+c)**(3/4),x)
 

Output:

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(3/4),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/4)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(3/4),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/4}} \,d x \] Input:

int(1/((a + b*x)^(3/2)*(c + d*x)^(3/4)),x)
 

Output:

int(1/((a + b*x)^(3/2)*(c + d*x)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx=\frac {4 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}-3 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a^{2} b \,d^{2}+2 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a \,b^{2} c d -3 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a \,b^{2} d^{2} x +2 \sqrt {d x +c}\, \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) b^{3} c d x}{\sqrt {d x +c}\, \left (3 a b d x -2 b^{2} c x +3 a^{2} d -2 a b c \right )} \] Input:

int(1/(b*x+a)^(3/2)/(d*x+c)^(3/4),x)
 

Output:

(4*(c + d*x)**(3/4)*sqrt(a + b*x) - 3*sqrt(c + d*x)*int(((c + d*x)**(1/4)* 
sqrt(a + b*x)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a**2*b*c* 
d*x + 6*a**2*b*d**2*x**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a*b**2*d* 
*2*x**3 - 2*b**3*c**2*x**2 - 2*b**3*c*d*x**3),x)*a**2*b*d**2 + 2*sqrt(c + 
d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 
2*a**2*b*c**2 + 4*a**2*b*c*d*x + 6*a**2*b*d**2*x**2 - 4*a*b**2*c**2*x - a* 
b**2*c*d*x**2 + 3*a*b**2*d**2*x**3 - 2*b**3*c**2*x**2 - 2*b**3*c*d*x**3),x 
)*a*b**2*c*d - 3*sqrt(c + d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(3*a 
**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a**2*b*c*d*x + 6*a**2*b*d**2*x 
**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a*b**2*d**2*x**3 - 2*b**3*c**2 
*x**2 - 2*b**3*c*d*x**3),x)*a*b**2*d**2*x + 2*sqrt(c + d*x)*int(((c + d*x) 
**(1/4)*sqrt(a + b*x)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a 
**2*b*c*d*x + 6*a**2*b*d**2*x**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a 
*b**2*d**2*x**3 - 2*b**3*c**2*x**2 - 2*b**3*c*d*x**3),x)*b**3*c*d*x)/(sqrt 
(c + d*x)*(3*a**2*d - 2*a*b*c + 3*a*b*d*x - 2*b**2*c*x))