3.17.61 \(\int \frac {\sqrt {a+b x}}{(c+d x)^{5/4}} \, dx\) [1661]

3.17.61.1 Optimal result

Integrand size = 19, antiderivative size = 190 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{5/4}} \, dx=-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}+\frac {8 \sqrt [4]{b} (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d^2 \sqrt {a+b x}}-\frac {8 \sqrt [4]{b} (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{d^2 \sqrt {a+b x}} \]

output

-4*(b*x+a)^(1/2)/d/(d*x+c)^(1/4)+8*b^(1/4)*(-a*d+b*c)^(3/4)*EllipticE(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)/d^2/(b 
*x+a)^(1/2)-8*b^(1/4)*(-a*d+b*c)^(3/4)*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)/d^2/(b*x+a)^(1/2)
 

3.17.61.2 Mathematica [C] (verified)

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 = 0.38 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{5/4}} \, dx=\frac {2 (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {3}{2},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (c+d x)^{5/4}} \]

input

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

output

(2*(a + b*x)^(3/2)*((b*(c + d*x))/(b*c - a*d))^(5/4)*Hypergeometric2F1[5/4 
, 3/2, 5/2, (d*(a + b*x))/(-(b*c) + a*d)])/(3*b*(c + d*x)^(5/4))
 

3.17.61.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {57, 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.

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {8 b \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {8 b \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {8 b \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {8 b \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {8 b \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {8 b \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {8 b \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {4 \sqrt {a+b x}}{d \sqrt [4]{c+d x}}\)

input

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

output

(-4*Sqrt[a + b*x])/(d*(c + d*x)^(1/4)) + (8*b*(((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])))/d^2
 

3.17.61.3.1 Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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])
 

3.17.61.4 Maple [F]

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

input

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

output

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

3.17.61.5 Fricas [F]

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

input

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

output

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

3.17.61.6 Sympy [F]

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

input

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

output

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

3.17.61.7 Maxima [F]

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

input

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

output

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

3.17.61.8 Giac [F]

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

input

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

output

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

3.17.61.9 Mupad [F(-1)]

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

input

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

output

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

3.17.61.10 Reduce [F]

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

input

int(sqrt(a + b*x)/((c + d*x)**(1/4)*(c + d*x)),x)
 

output

int(sqrt(a + b*x)/((c + d*x)**(1/4)*c + (c + d*x)**(1/4)*d*x),x)