\(\int \frac {1}{(a+b x)^{5/2} \sqrt [4]{c+d x}} \, dx\) [456]

Optimal result

Integrand size = 19, antiderivative size = 147 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [4]{c+d x}} \, dx=-\frac {2}{3 b (a+b x)^{3/2} \sqrt [4]{c+d x}}+\frac {d}{3 b (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}+\frac {d^{3/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{b (b c-a d)^{3/2} \sqrt [4]{c+d x}} \] Output:

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

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 = 0.50 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [4]{c+d x}} \, dx=-\frac {2 \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \sqrt [4]{c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {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.

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*(c + d*x)^(3/4))/(3*(b*c - a*d)*(a + b*x)^(3/2)) - (d*((-2*(c + d*x)^( 
3/4))/((b*c - a*d)*Sqrt[a + b*x]) + (2*(((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])))/(b*c - a*d)))/(2*(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 327

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]
 

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]
 

rule 836

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]
 

rule 1388

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]))
 

rule 1390

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x + a)*(d*x + c)^(3/4)/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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