\(\int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx\) [462]

Optimal result

Integrand size = 19, antiderivative size = 149 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=-\frac {2 \sqrt [4]{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {5 d \sqrt [4]{c+d x}}{3 (b c-a d)^2 \sqrt {a+b x}}+\frac {5 d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt {a+b x}} \] Output:

-2/3*(d*x+c)^(1/4)/(-a*d+b*c)/(b*x+a)^(3/2)+5/3*d*(d*x+c)^(1/4)/(-a*d+b*c) 
^2/(b*x+a)^(1/2)+5/3*d*(-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)^(7/4)/(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 = 73, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} (c+d x)^{3/4}} \] Input:

Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(3/4)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.

\(\displaystyle \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {5 d \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}}dx}{6 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {5 d \left (-\frac {d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}}dx}{2 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\right )}{6 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {5 d \left (-\frac {2 \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{b c-a d}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\right )}{6 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 765

\(\displaystyle -\frac {5 d \left (-\frac {2 \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\right )}{6 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 762

\(\displaystyle -\frac {5 d \left (-\frac {2 \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt [4]{b} (b c-a d)^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}\right )}{6 (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\)

Input:

Int[1/((a + b*x)^(5/2)*(c + d*x)^(3/4)),x]
 

Output:

(-2*(c + d*x)^(1/4))/(3*(b*c - a*d)*(a + b*x)^(3/2)) - (5*d*((-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])))/(6*(b*c - a 
*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]

\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (x d +c \right )^{\frac {3}{4}}}d x\]

Input:

int(1/(b*x+a)^(5/2)/(d*x+c)^(3/4),x)
 

Output:

int(1/(b*x+a)^(5/2)/(d*x+c)^(3/4),x)
 

Fricas [F]

\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(3/4),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(1/(b*x+a)**(5/2)/(d*x+c)**(3/4),x)
 

Output:

Integral(1/((a + b*x)**(5/2)*(c + d*x)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(3/4),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/4)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(3/4),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/4}} \,d x \] Input:

int(1/((a + b*x)^(5/2)*(c + d*x)^(3/4)),x)
 

Output:

int(1/((a + b*x)^(5/2)*(c + d*x)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx =\text {Too large to display} \] Input:

int(1/(b*x+a)^(5/2)/(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)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c*d*x + 
3*a**3*b*d**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2*b* 
*2*d**2*x**3 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 - 
 2*b**4*c**2*x**3 - 2*b**4*c*d*x**4),x)*a**3*b*d**2 - 6*sqrt(c + d*x)*int( 
((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 
 + a**3*b*c*d*x + 3*a**3*b*d**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c* 
d*x**2 + 3*a**2*b**2*d**2*x**3 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + 
a*b**3*d**2*x**4 - 2*b**4*c**2*x**3 - 2*b**4*c*d*x**4),x)*a**2*b**2*c*d + 
6*sqrt(c + d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**4*c*d + a**4*d* 
*2*x - 2*a**3*b*c**2 + a**3*b*c*d*x + 3*a**3*b*d**2*x**2 - 6*a**2*b**2*c** 
2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2*b**2*d**2*x**3 - 6*a*b**3*c**2*x**2 - 
5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 - 2*b**4*c**2*x**3 - 2*b**4*c*d*x**4) 
,x)*a**2*b**2*d**2*x - 12*sqrt(c + d*x)*int(((c + d*x)**(1/4)*sqrt(a + b*x 
)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c*d*x + 3*a**3*b*d** 
2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2*b**2*d**2*x**3 
 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 - 2*b**4*c**2 
*x**3 - 2*b**4*c*d*x**4),x)*a*b**3*c*d*x + 3*sqrt(c + d*x)*int(((c + d*x)* 
*(1/4)*sqrt(a + b*x)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c 
*d*x + 3*a**3*b*d**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 +...