\(\int \sqrt {a-b x} \sqrt [6]{a+b x} \, dx\) [291]

Optimal result

Integrand size = 20, antiderivative size = 344 \[ \int \sqrt {a-b x} \sqrt [6]{a+b x} \, dx=\frac {3 a \sqrt {a-b x} \sqrt [6]{a+b x}}{10 b}-\frac {3 (a-b x)^{3/2} \sqrt [6]{a+b x}}{5 b}+\frac {3\ 3^{3/4} a^{5/3} \sqrt [6]{a+b x} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{10 \sqrt [3]{2} b \sqrt {a-b x} \sqrt {-\frac {\sqrt [3]{a+b x} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x}\right )^2}}} \] Output:

3/10*a*(-b*x+a)^(1/2)*(b*x+a)^(1/6)/b-3/5*(-b*x+a)^(3/2)*(b*x+a)^(1/6)/b+3 
/20*3^(3/4)*a^(5/3)*(b*x+a)^(1/6)*(2^(1/3)*a^(1/3)-(b*x+a)^(1/3))*((2^(2/3 
)*a^(2/3)+2^(1/3)*a^(1/3)*(b*x+a)^(1/3)+(b*x+a)^(2/3))/(2^(1/3)*a^(1/3)-(1 
+3^(1/2))*(b*x+a)^(1/3))^2)^(1/2)*InverseJacobiAM(arccos((2^(1/3)*a^(1/3)- 
(1-3^(1/2))*(b*x+a)^(1/3))/(2^(1/3)*a^(1/3)-(1+3^(1/2))*(b*x+a)^(1/3))),1/ 
4*6^(1/2)+1/4*2^(1/2))*2^(2/3)/b/(-b*x+a)^(1/2)/(-(b*x+a)^(1/3)*(2^(1/3)*a 
^(1/3)-(b*x+a)^(1/3))/(2^(1/3)*a^(1/3)-(1+3^(1/2))*(b*x+a)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.19 \[ \int \sqrt {a-b x} \sqrt [6]{a+b x} \, dx=-\frac {2 \sqrt [6]{2} (a-b x)^{3/2} \sqrt [6]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{2},\frac {5}{2},\frac {a-b x}{2 a}\right )}{3 b \sqrt [6]{\frac {a+b x}{a}}} \] Input:

Integrate[Sqrt[a - b*x]*(a + b*x)^(1/6),x]
 

Output:

(-2*2^(1/6)*(a - b*x)^(3/2)*(a + b*x)^(1/6)*Hypergeometric2F1[-1/6, 3/2, 5 
/2, (a - b*x)/(2*a)])/(3*b*((a + b*x)/a)^(1/6))
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {60, 60, 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.

\(\displaystyle \int \sqrt {a-b x} \sqrt [6]{a+b x} \, dx\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{5} a \left (\frac {9 a \int \frac {1}{\sqrt {a-b x}}d\sqrt [6]{a+b x}}{b}+\frac {3 \sqrt {a-b x} \sqrt [6]{a+b x}}{2 b}\right )-\frac {3 (a-b x)^{3/2} \sqrt [6]{a+b x}}{5 b}\)

\(\Big \downarrow \) 766

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

Input:

Int[Sqrt[a - b*x]*(a + b*x)^(1/6),x]
 

Output:

(-3*(a - b*x)^(3/2)*(a + b*x)^(1/6))/(5*b) + (a*((3*Sqrt[a - b*x]*(a + b*x 
)^(1/6))/(2*b) + (3*3^(3/4)*a^(2/3)*(a + b*x)^(1/6)*(2^(1/3)*a^(1/3) - (a 
+ b*x)^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x)^(1/3) + (a 
 + b*x)^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + b*x)^(1/3))^2]*Ellipt 
icF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + b*x)^(1/3))/(2^(1/3)*a^(1 
/3) - (1 + Sqrt[3])*(a + b*x)^(1/3))], (2 + Sqrt[3])/4])/(2*2^(1/3)*b*Sqrt 
[a - b*x]*Sqrt[-(((a + b*x)^(1/3)*(2^(1/3)*a^(1/3) - (a + b*x)^(1/3)))/(2^ 
(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + b*x)^(1/3))^2)])))/5
 

Defintions of rubi rules used

rule 60

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

Maple [F]

\[\int \sqrt {-b x +a}\, \left (b x +a \right )^{\frac {1}{6}}d x\]

Input:

int((-b*x+a)^(1/2)*(b*x+a)^(1/6),x)
 

Output:

int((-b*x+a)^(1/2)*(b*x+a)^(1/6),x)
 

Fricas [F]

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

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

Output:

integral((b*x + a)^(1/6)*sqrt(-b*x + a), x)
 

Sympy [F]

\[ \int \sqrt {a-b x} \sqrt [6]{a+b x} \, dx=\int \sqrt {a - b x} \sqrt [6]{a + b x}\, dx \] Input:

integrate((-b*x+a)**(1/2)*(b*x+a)**(1/6),x)
 

Output:

Integral(sqrt(a - b*x)*(a + b*x)**(1/6), x)
 

Maxima [F]

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

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

Output:

integrate((b*x + a)^(1/6)*sqrt(-b*x + a), x)
 

Giac [F]

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

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

Output:

integrate((b*x + a)^(1/6)*sqrt(-b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a-b x} \sqrt [6]{a+b x} \, dx=\int {\left (a+b\,x\right )}^{1/6}\,\sqrt {a-b\,x} \,d x \] Input:

int((a + b*x)^(1/6)*(a - b*x)^(1/2),x)
 

Output:

int((a + b*x)^(1/6)*(a - b*x)^(1/2), x)
 

Reduce [F]

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

int((-b*x+a)^(1/2)*(b*x+a)^(1/6),x)
 

Output:

int((a + b*x)**(1/6)*sqrt(a - b*x),x)