\(\int \frac {(c x)^{3/2}}{\sqrt [4]{a+b x^2}} \, dx\) [1028]

Optimal result

Integrand size = 19, antiderivative size = 117 \[ \int \frac {(c x)^{3/2}}{\sqrt [4]{a+b x^2}} \, dx=\frac {c \sqrt {c x} \left (a+b x^2\right )^{3/4}}{2 b}-\frac {a c^{3/2} \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{4 b^{5/4}}-\frac {a c^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{4 b^{5/4}} \] Output:

1/2*c*(c*x)^(1/2)*(b*x^2+a)^(3/4)/b-1/4*a*c^(3/2)*arctan(b^(1/4)*(c*x)^(1/ 
2)/c^(1/2)/(b*x^2+a)^(1/4))/b^(5/4)-1/4*a*c^(3/2)*arctanh(b^(1/4)*(c*x)^(1 
/2)/c^(1/2)/(b*x^2+a)^(1/4))/b^(5/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83 \[ \int \frac {(c x)^{3/2}}{\sqrt [4]{a+b x^2}} \, dx=\frac {(c x)^{3/2} \left (2 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^{3/4}-a \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )-a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{4 b^{5/4} x^{3/2}} \] Input:

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

Output:

((c*x)^(3/2)*(2*b^(1/4)*Sqrt[x]*(a + b*x^2)^(3/4) - a*ArcTan[(b^(1/4)*Sqrt 
[x])/(a + b*x^2)^(1/4)] - a*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)])) 
/(4*b^(5/4)*x^(3/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {262, 266, 770, 756, 218, 221}

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.

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(c*Sqrt[c*x]*(a + b*x^2)^(3/4))/(2*b) - (a*c*((Sqrt[c]*ArcTan[(b^(1/4)*Sqr 
t[c*x])/(Sqrt[c]*(a + b*x^2)^(1/4))])/(2*b^(1/4)) + (Sqrt[c]*ArcTanh[(b^(1 
/4)*Sqrt[c*x])/(Sqrt[c]*(a + b*x^2)^(1/4))])/(2*b^(1/4))))/(2*b)
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

Maple [F]

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

Input:

int((c*x)^(3/2)/(b*x^2+a)^(1/4),x)
 

Output:

int((c*x)^(3/2)/(b*x^2+a)^(1/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.57 \[ \int \frac {(c x)^{3/2}}{\sqrt [4]{a+b x^2}} \, dx=\frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} c - \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} a c + \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} {\left (b^{2} x^{2} + a b\right )}}{b x^{2} + a}\right ) + \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} a c - \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} {\left (b^{2} x^{2} + a b\right )}}{b x^{2} + a}\right ) + i \, \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} a c - \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} {\left (i \, b^{2} x^{2} + i \, a b\right )}}{b x^{2} + a}\right ) - i \, \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} a c - \left (\frac {a^{4} c^{6}}{b^{5}}\right )^{\frac {1}{4}} {\left (-i \, b^{2} x^{2} - i \, a b\right )}}{b x^{2} + a}\right )}{8 \, b} \] Input:

integrate((c*x)^(3/2)/(b*x^2+a)^(1/4),x, algorithm="fricas")
 

Output:

1/8*(4*(b*x^2 + a)^(3/4)*sqrt(c*x)*c - (a^4*c^6/b^5)^(1/4)*b*log(((b*x^2 + 
 a)^(3/4)*sqrt(c*x)*a*c + (a^4*c^6/b^5)^(1/4)*(b^2*x^2 + a*b))/(b*x^2 + a) 
) + (a^4*c^6/b^5)^(1/4)*b*log(((b*x^2 + a)^(3/4)*sqrt(c*x)*a*c - (a^4*c^6/ 
b^5)^(1/4)*(b^2*x^2 + a*b))/(b*x^2 + a)) + I*(a^4*c^6/b^5)^(1/4)*b*log(((b 
*x^2 + a)^(3/4)*sqrt(c*x)*a*c - (a^4*c^6/b^5)^(1/4)*(I*b^2*x^2 + I*a*b))/( 
b*x^2 + a)) - I*(a^4*c^6/b^5)^(1/4)*b*log(((b*x^2 + a)^(3/4)*sqrt(c*x)*a*c 
 - (a^4*c^6/b^5)^(1/4)*(-I*b^2*x^2 - I*a*b))/(b*x^2 + a)))/b
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.38 \[ \int \frac {(c x)^{3/2}}{\sqrt [4]{a+b x^2}} \, dx=\frac {c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((c*x)**(3/2)/(b*x**2+a)**(1/4),x)
 

Output:

c**(3/2)*x**(5/2)*gamma(5/4)*hyper((1/4, 5/4), (9/4,), b*x**2*exp_polar(I* 
pi)/a)/(2*a**(1/4)*gamma(9/4))
 

Maxima [F]

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

integrate((c*x)^(3/2)/(b*x^2+a)^(1/4),x, algorithm="maxima")
 

Output:

integrate((c*x)^(3/2)/(b*x^2 + a)^(1/4), x)
 

Giac [F]

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

integrate((c*x)^(3/2)/(b*x^2+a)^(1/4),x, algorithm="giac")
 

Output:

integrate((c*x)^(3/2)/(b*x^2 + a)^(1/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{3/2}}{\sqrt [4]{a+b x^2}} \, dx=\int \frac {{\left (c\,x\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{1/4}} \,d x \] Input:

int((c*x)^(3/2)/(a + b*x^2)^(1/4),x)
 

Output:

int((c*x)^(3/2)/(a + b*x^2)^(1/4), x)
 

Reduce [F]

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

int((c*x)^(3/2)/(b*x^2+a)^(1/4),x)
 

Output:

sqrt(c)*int((sqrt(x)*x)/(a + b*x**2)**(1/4),x)*c