\(\int \frac {1}{(a+b x^{4/3})^{3/2}} \, dx\) [145]

Optimal result

Integrand size = 13, antiderivative size = 271 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^{3/2}} \, dx=\frac {3 x}{2 a \sqrt {a+b x^{4/3}}}-\frac {3 \sqrt [3]{x} \sqrt {a+b x^{4/3}}}{2 a \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^{2/3}\right )}+\frac {3 \left (\sqrt {a}+\sqrt {b} x^{2/3}\right ) \sqrt {\frac {a+b x^{4/3}}{\left (\sqrt {a}+\sqrt {b} x^{2/3}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt {a+b x^{4/3}}}-\frac {3 \left (\sqrt {a}+\sqrt {b} x^{2/3}\right ) \sqrt {\frac {a+b x^{4/3}}{\left (\sqrt {a}+\sqrt {b} x^{2/3}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{3/4} b^{3/4} \sqrt {a+b x^{4/3}}} \] Output:

3/2*x/a/(a+b*x^(4/3))^(1/2)-3/2*x^(1/3)*(a+b*x^(4/3))^(1/2)/a/b^(1/2)/(a^( 
1/2)+b^(1/2)*x^(2/3))+3/2*(a^(1/2)+b^(1/2)*x^(2/3))*((a+b*x^(4/3))/(a^(1/2 
)+b^(1/2)*x^(2/3))^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*x^(1/3)/a^(1/4) 
)),1/2*2^(1/2))/a^(3/4)/b^(3/4)/(a+b*x^(4/3))^(1/2)-3/4*(a^(1/2)+b^(1/2)*x 
^(2/3))*((a+b*x^(4/3))/(a^(1/2)+b^(1/2)*x^(2/3))^2)^(1/2)*InverseJacobiAM( 
2*arctan(b^(1/4)*x^(1/3)/a^(1/4)),1/2*2^(1/2))/a^(3/4)/b^(3/4)/(a+b*x^(4/3 
))^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.20 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^{3/2}} \, dx=\frac {x \sqrt {1+\frac {b x^{4/3}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^{4/3}}{a}\right )}{a \sqrt {a+b x^{4/3}}} \] Input:

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

Output:

(x*Sqrt[1 + (b*x^(4/3))/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^(4/3))/ 
a)])/(a*Sqrt[a + b*x^(4/3)])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {774, 819, 834, 27, 761, 1510}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^{4/3}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 774

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

\(\Big \downarrow \) 819

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

3*(x/(2*a*Sqrt[a + b*x^(4/3)]) - (-((-((x^(1/3)*Sqrt[a + b*x^(4/3)])/(Sqrt 
[a] + Sqrt[b]*x^(2/3))) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x^(2/3))*Sqrt[(a + b 
*x^(4/3))/(Sqrt[a] + Sqrt[b]*x^(2/3))^2]*EllipticE[2*ArcTan[(b^(1/4)*x^(1/ 
3))/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^(4/3)]))/Sqrt[b]) + (a^(1/4)*(Sq 
rt[a] + Sqrt[b]*x^(2/3))*Sqrt[(a + b*x^(4/3))/(Sqrt[a] + Sqrt[b]*x^(2/3))^ 
2]*EllipticF[2*ArcTan[(b^(1/4)*x^(1/3))/a^(1/4)], 1/2])/(2*b^(3/4)*Sqrt[a 
+ b*x^(4/3)]))/(2*a))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 774

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, 
Simp[k   Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre 
eQ[{a, b, p}, x] && FractionQ[n]
 

rule 819

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

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.45

method	result	size
derivativedivides	\(\frac {3 x}{2 a \sqrt {\left (x^{\frac {4}{3}}+\frac {a}{b}\right ) b}}-\frac {3 i \sqrt {1-\frac {i \sqrt {b}\, x^{\frac {2}{3}}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{\frac {2}{3}}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x^{\frac {1}{3}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x^{\frac {1}{3}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \,x^{\frac {4}{3}}}\, \sqrt {b}}\)	\(121\)
default	\(\frac {3 x}{2 a \sqrt {\left (x^{\frac {4}{3}}+\frac {a}{b}\right ) b}}-\frac {3 i \sqrt {1-\frac {i \sqrt {b}\, x^{\frac {2}{3}}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{\frac {2}{3}}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x^{\frac {1}{3}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x^{\frac {1}{3}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \,x^{\frac {4}{3}}}\, \sqrt {b}}\)	\(121\)

Input:

int(1/(a+b*x^(4/3))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

3/2/a*x/((x^(4/3)+a/b)*b)^(1/2)-3/2*I/a^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1 
-I/a^(1/2)*b^(1/2)*x^(2/3))^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^(2/3))^(1/2)/(a+b 
*x^(4/3))^(1/2)/b^(1/2)*(EllipticF(x^(1/3)*(I/a^(1/2)*b^(1/2))^(1/2),I)-El 
lipticE(x^(1/3)*(I/a^(1/2)*b^(1/2))^(1/2),I))
 

Fricas [F]

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

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

Output:

integral(-(2*a*b^3*x^4 - 3*a^2*b^2*x^(8/3) - a^4 - (b^4*x^5 - 2*a^3*b*x)*x 
^(1/3))*sqrt(b*x^(4/3) + a)/(b^6*x^8 + 2*a^3*b^3*x^4 + a^6), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^{3/2}} \, dx=\frac {3 x \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{\frac {4}{3}} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

3*x*gamma(3/4)*hyper((3/4, 3/2), (7/4,), b*x**(4/3)*exp_polar(I*pi)/a)/(4* 
a**(3/2)*gamma(7/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^{3/2}} \, dx=\frac {x\,{\left (\frac {b\,x^{4/3}}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {3}{2};\ \frac {7}{4};\ -\frac {b\,x^{4/3}}{a}\right )}{{\left (a+b\,x^{4/3}\right )}^{3/2}} \] Input:

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

Output:

(x*((b*x^(4/3))/a + 1)^(3/2)*hypergeom([3/4, 3/2], 7/4, -(b*x^(4/3))/a))/( 
a + b*x^(4/3))^(3/2)
 

Reduce [F]

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

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

Output:

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