\(\int \frac {x}{\sqrt {c+d x} \sqrt [3]{a+b x^2}} \, dx\) [1636]

Optimal result

Integrand size = 22, antiderivative size = 180 \[ \int \frac {x}{\sqrt {c+d x} \sqrt [3]{a+b x^2}} \, dx=\frac {3 \left (a+b x^2\right )^{2/3}}{4 b \sqrt {c+d x}}-\frac {3 \left (a+b x^2\right )^{2/3} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {2}{3},-\frac {2}{3},\frac {1}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{4 b \sqrt {c+d x} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{2/3} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{2/3}} \] Output:

3/4*(b*x^2+a)^(2/3)/b/(d*x+c)^(1/2)-3/4*(b*x^2+a)^(2/3)*AppellF1(-1/2,-2/3 
,-2/3,1/2,(d*x+c)/(c-(-a)^(1/2)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2) 
))/b/(d*x+c)^(1/2)/(1-(d*x+c)/(c-(-a)^(1/2)*d/b^(1/2)))^(2/3)/(1-(d*x+c)/( 
c+(-a)^(1/2)*d/b^(1/2)))^(2/3)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(513\) vs. \(2(180)=360\).

Time = 22.41 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.85 \[ \int \frac {x}{\sqrt {c+d x} \sqrt [3]{a+b x^2}} \, dx=\frac {6 \left (7 d^2 \left (a+b x^2\right )+21 b c (c+d x) \sqrt [3]{\frac {a d^2+c \sqrt {-a b d^2}-b c d x+d \sqrt {-a b d^2} x}{\left (-b c+\sqrt {-a b d^2}\right ) (c+d x)}} \sqrt [3]{\frac {-a d^2+c \sqrt {-a b d^2}+b c d x+d \sqrt {-a b d^2} x}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},\frac {1}{3},\frac {7}{6},\frac {b c^2+a d^2}{\left (b c-\sqrt {-a b d^2}\right ) (c+d x)},\frac {b c^2+a d^2}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )-3 \left (b c^2+a d^2\right ) \sqrt [3]{\frac {a d^2+c \sqrt {-a b d^2}-b c d x+d \sqrt {-a b d^2} x}{\left (-b c+\sqrt {-a b d^2}\right ) (c+d x)}} \sqrt [3]{\frac {-a d^2+c \sqrt {-a b d^2}+b c d x+d \sqrt {-a b d^2} x}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}} \operatorname {AppellF1}\left (\frac {7}{6},\frac {1}{3},\frac {1}{3},\frac {13}{6},\frac {b c^2+a d^2}{\left (b c-\sqrt {-a b d^2}\right ) (c+d x)},\frac {b c^2+a d^2}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )\right )}{35 b d^2 \sqrt {c+d x} \sqrt [3]{a+b x^2}} \] Input:

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

Output:

(6*(7*d^2*(a + b*x^2) + 21*b*c*(c + d*x)*((a*d^2 + c*Sqrt[-(a*b*d^2)] - b* 
c*d*x + d*Sqrt[-(a*b*d^2)]*x)/((-(b*c) + Sqrt[-(a*b*d^2)])*(c + d*x)))^(1/ 
3)*((-(a*d^2) + c*Sqrt[-(a*b*d^2)] + b*c*d*x + d*Sqrt[-(a*b*d^2)]*x)/((b*c 
 + Sqrt[-(a*b*d^2)])*(c + d*x)))^(1/3)*AppellF1[1/6, 1/3, 1/3, 7/6, (b*c^2 
 + a*d^2)/((b*c - Sqrt[-(a*b*d^2)])*(c + d*x)), (b*c^2 + a*d^2)/((b*c + Sq 
rt[-(a*b*d^2)])*(c + d*x))] - 3*(b*c^2 + a*d^2)*((a*d^2 + c*Sqrt[-(a*b*d^2 
)] - b*c*d*x + d*Sqrt[-(a*b*d^2)]*x)/((-(b*c) + Sqrt[-(a*b*d^2)])*(c + d*x 
)))^(1/3)*((-(a*d^2) + c*Sqrt[-(a*b*d^2)] + b*c*d*x + d*Sqrt[-(a*b*d^2)]*x 
)/((b*c + Sqrt[-(a*b*d^2)])*(c + d*x)))^(1/3)*AppellF1[7/6, 1/3, 1/3, 13/6 
, (b*c^2 + a*d^2)/((b*c - Sqrt[-(a*b*d^2)])*(c + d*x)), (b*c^2 + a*d^2)/(( 
b*c + Sqrt[-(a*b*d^2)])*(c + d*x))]))/(35*b*d^2*Sqrt[c + d*x]*(a + b*x^2)^ 
(1/3))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.69, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {624, 514, 150}

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.

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

\(\Big \downarrow \) 624

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {2 (c+d x)^{3/2} \sqrt [3]{1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt [3]{1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},\frac {1}{3},\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 d^2 \sqrt [3]{a+b x^2}}-\frac {2 c \sqrt {c+d x} \sqrt [3]{1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt [3]{1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 \sqrt [3]{a+b x^2}}\)

Input:

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

Output:

(-2*c*Sqrt[c + d*x]*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^(1/3)*(1 - 
(c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^(1/3)*AppellF1[1/2, 1/3, 1/3, 3/2, ( 
c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])] 
)/(d^2*(a + b*x^2)^(1/3)) + (2*(c + d*x)^(3/2)*(1 - (c + d*x)/(c - (Sqrt[- 
a]*d)/Sqrt[b]))^(1/3)*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^(1/3)*App 
ellF1[3/2, 1/3, 1/3, 5/2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/ 
(c + (Sqrt[-a]*d)/Sqrt[b])])/(3*d^2*(a + b*x^2)^(1/3))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 624

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[1/d   Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si 
mp[c/d   Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x}{\sqrt {c+d x} \sqrt [3]{a+b x^2}} \, dx=\int \frac {x}{\sqrt [3]{a + b x^{2}} \sqrt {c + d x}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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