3.18.40 \(\int \frac {\sqrt [6]{c+d x}}{(a+b x)^{3/2}} \, dx\) [1740]

3.18.40.1 Optimal result

Integrand size = 19, antiderivative size = 367 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \sqrt [6]{c+d x}}{b \sqrt {a+b x}}+\frac {\sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} b \sqrt [3]{b c-a d} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

output

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

3.18.40.2 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 = 0.19 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \sqrt [6]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{6},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

input

Integrate[(c + d*x)^(1/6)/(a + b*x)^(3/2),x]
 

output

(-2*(c + d*x)^(1/6)*Hypergeometric2F1[-1/2, -1/6, 1/2, (d*(a + b*x))/(-(b* 
c) + a*d)])/(b*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(1/6))
 

3.18.40.3 Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {57, 73, 766}

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.

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{b}-\frac {2 \sqrt [6]{c+d x}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 766

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

input

Int[(c + d*x)^(1/6)/(a + b*x)^(3/2),x]
 

output

(-2*(c + d*x)^(1/6))/(b*Sqrt[a + b*x]) + ((c + d*x)^(1/6)*((b*c - a*d)^(1/ 
3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d 
)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 
 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcCos[((b*c - a*d)^(1/3 
) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[ 
3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(3^(1/4)*b*(b*c - a*d)^(1 
/3)*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x) 
^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]*S 
qrt[a - (b*c)/d + (b*(c + d*x))/d])
 

3.18.40.3.1 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 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

3.18.40.4 Maple [F]

\[\int \frac {\left (d x +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]

input

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

output

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

3.18.40.5 Fricas [F]

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

input

integrate((d*x+c)^(1/6)/(b*x+a)^(3/2),x, algorithm="fricas")
 

output

integral(sqrt(b*x + a)*(d*x + c)^(1/6)/(b^2*x^2 + 2*a*b*x + a^2), x)
 

3.18.40.6 Sympy [F]

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

input

integrate((d*x+c)**(1/6)/(b*x+a)**(3/2),x)
 

output

Integral((c + d*x)**(1/6)/(a + b*x)**(3/2), x)
 

3.18.40.7 Maxima [F]

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

input

integrate((d*x+c)^(1/6)/(b*x+a)^(3/2),x, algorithm="maxima")
 

output

integrate((d*x + c)^(1/6)/(b*x + a)^(3/2), x)
 

3.18.40.8 Giac [F]

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

input

integrate((d*x+c)^(1/6)/(b*x+a)^(3/2),x, algorithm="giac")
 

output

integrate((d*x + c)^(1/6)/(b*x + a)^(3/2), x)
 

3.18.40.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/6}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]

input

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

output

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