\(\int (c x)^{5/4} \sqrt {a+b x^2} \, dx\) [665]

Optimal result

Integrand size = 19, antiderivative size = 537 \[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\frac {16 a c \sqrt [4]{c x} \sqrt {a+b x^2}}{65 b}+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}-\frac {8 a^2 c (c x)^{3/4} \sqrt {-\frac {a+b x^2}{\sqrt {a} \sqrt {b} x}} \sqrt {\frac {\left (\sqrt [4]{a} \sqrt {c}+\sqrt [4]{b} \sqrt {c x}\right )^2}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {2}+\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a}}-\frac {2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt [4]{b} \sqrt {c x}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{65 \sqrt {2+\sqrt {2}} b^{3/4} \sqrt {a+b x^2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt [4]{b} \sqrt {c x}\right )}+\frac {8 a^2 c (c x)^{3/4} \sqrt {-\frac {a+b x^2}{\sqrt {a} \sqrt {b} x}} \sqrt {-\frac {\left (\sqrt [4]{a} \sqrt {c}-\sqrt [4]{b} \sqrt {c x}\right )^2}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {2}+\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a}}+\frac {2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt [4]{b} \sqrt {c x}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{65 \sqrt {2+\sqrt {2}} b^{3/4} \sqrt {a+b x^2} \left (\sqrt [4]{a} \sqrt {c}-\sqrt [4]{b} \sqrt {c x}\right )} \] Output:

16/65*a*c*(c*x)^(1/4)*(b*x^2+a)^(1/2)/b+4/13*(c*x)^(9/4)*(b*x^2+a)^(1/2)/c 
-8/65*a^2*c*(c*x)^(3/4)*(-(b*x^2+a)/a^(1/2)/b^(1/2)/x)^(1/2)*((a^(1/4)*c^( 
1/2)+b^(1/4)*(c*x)^(1/2))^2/a^(1/4)/b^(1/4)/c^(1/2)/(c*x)^(1/2))^(1/2)*Ell 
ipticF(1/2*(-a^(1/4)*c^(1/2)*(2^(1/2)+2^(1/2)*b^(1/2)*x/a^(1/2)-2*b^(1/4)* 
(c*x)^(1/2)/a^(1/4)/c^(1/2))/b^(1/4)/(c*x)^(1/2))^(1/2),(-2+2*2^(1/2))^(1/ 
2))/(2+2^(1/2))^(1/2)/b^(3/4)/(b*x^2+a)^(1/2)/(a^(1/4)*c^(1/2)+b^(1/4)*(c* 
x)^(1/2))+8/65*a^2*c*(c*x)^(3/4)*(-(b*x^2+a)/a^(1/2)/b^(1/2)/x)^(1/2)*(-(a 
^(1/4)*c^(1/2)-b^(1/4)*(c*x)^(1/2))^2/a^(1/4)/b^(1/4)/c^(1/2)/(c*x)^(1/2)) 
^(1/2)*EllipticF(1/2*(a^(1/4)*c^(1/2)*(2^(1/2)+2^(1/2)*b^(1/2)*x/a^(1/2)+2 
*b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))/b^(1/4)/(c*x)^(1/2))^(1/2),(-2+2*2^( 
1/2))^(1/2))/(2+2^(1/2))^(1/2)/b^(3/4)/(b*x^2+a)^(1/2)/(a^(1/4)*c^(1/2)-b^ 
(1/4)*(c*x)^(1/2))
 

Mathematica [C] (verified)

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

Time = 10.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.16 \[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\frac {4 c \sqrt [4]{c x} \sqrt {a+b x^2} \left (\left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{8},\frac {9}{8},-\frac {b x^2}{a}\right )\right )}{13 b \sqrt {1+\frac {b x^2}{a}}} \] Input:

Integrate[(c*x)^(5/4)*Sqrt[a + b*x^2],x]
 

Output:

(4*c*(c*x)^(1/4)*Sqrt[a + b*x^2]*((a + b*x^2)*Sqrt[1 + (b*x^2)/a] - a*Hype 
rgeometric2F1[-1/2, 1/8, 9/8, -((b*x^2)/a)]))/(13*b*Sqrt[1 + (b*x^2)/a])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {248, 262, 266, 767, 27, 2422}

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.

\(\displaystyle \int (c x)^{5/4} \sqrt {a+b x^2} \, dx\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {4}{13} a \int \frac {(c x)^{5/4}}{\sqrt {b x^2+a}}dx+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {4}{13} a \left (\frac {4 c \sqrt [4]{c x} \sqrt {a+b x^2}}{5 b}-\frac {a c^2 \int \frac {1}{(c x)^{3/4} \sqrt {b x^2+a}}dx}{5 b}\right )+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {4}{13} a \left (\frac {4 c \sqrt [4]{c x} \sqrt {a+b x^2}}{5 b}-\frac {4 a c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt [4]{c x}}{5 b}\right )+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}\)

\(\Big \downarrow \) 767

\(\displaystyle \frac {4}{13} a \left (\frac {4 c \sqrt [4]{c x} \sqrt {a+b x^2}}{5 b}-\frac {4 a c \left (\frac {1}{2} \int \frac {\sqrt [4]{a} \sqrt {c}-\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c} \sqrt {b x^2+a}}d\sqrt [4]{c x}+\frac {1}{2} \int \frac {\sqrt [4]{a} \sqrt {c}+\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c} \sqrt {b x^2+a}}d\sqrt [4]{c x}\right )}{5 b}\right )+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{13} a \left (\frac {4 c \sqrt [4]{c x} \sqrt {a+b x^2}}{5 b}-\frac {4 a c \left (\frac {\int \frac {\sqrt [4]{a} \sqrt {c}-\sqrt [4]{b} \sqrt {c x}}{\sqrt {b x^2+a}}d\sqrt [4]{c x}}{2 \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {c}+\sqrt [4]{b} \sqrt {c x}}{\sqrt {b x^2+a}}d\sqrt [4]{c x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{5 b}\right )+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}\)

\(\Big \downarrow \) 2422

\(\displaystyle \frac {4}{13} a \left (\frac {4 c \sqrt [4]{c x} \sqrt {a+b x^2}}{5 b}-\frac {4 a c \left (\frac {\sqrt [4]{b} (c x)^{3/4} \sqrt {-\frac {a c^2+b c^2 x^2}{\sqrt {a} \sqrt {b} c^2 x}} \sqrt {\frac {\left (\sqrt [4]{a} \sqrt {c}+\sqrt [4]{b} \sqrt {c x}\right )^2}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} \sqrt {b} x c+\sqrt {2} \sqrt {a} c-2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x} \sqrt {c}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {a+b x^2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt [4]{b} \sqrt {c x}\right )}-\frac {\sqrt [4]{b} (c x)^{3/4} \sqrt {-\frac {a c^2+b c^2 x^2}{\sqrt {a} \sqrt {b} c^2 x}} \sqrt {-\frac {\left (\sqrt [4]{a} \sqrt {c}-\sqrt [4]{b} \sqrt {c x}\right )^2}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2} \sqrt {b} x c+\sqrt {2} \sqrt {a} c+2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x} \sqrt {c}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {a+b x^2} \left (\sqrt [4]{a} \sqrt {c}-\sqrt [4]{b} \sqrt {c x}\right )}\right )}{5 b}\right )+\frac {4 (c x)^{9/4} \sqrt {a+b x^2}}{13 c}\)

Input:

Int[(c*x)^(5/4)*Sqrt[a + b*x^2],x]
 

Output:

(4*(c*x)^(9/4)*Sqrt[a + b*x^2])/(13*c) + (4*a*((4*c*(c*x)^(1/4)*Sqrt[a + b 
*x^2])/(5*b) - (4*a*c*((b^(1/4)*(c*x)^(3/4)*Sqrt[-((a*c^2 + b*c^2*x^2)/(Sq 
rt[a]*Sqrt[b]*c^2*x))]*Sqrt[(a^(1/4)*Sqrt[c] + b^(1/4)*Sqrt[c*x])^2/(a^(1/ 
4)*b^(1/4)*Sqrt[c]*Sqrt[c*x])]*EllipticF[ArcSin[Sqrt[-((Sqrt[2]*Sqrt[a]*c 
+ Sqrt[2]*Sqrt[b]*c*x - 2*a^(1/4)*b^(1/4)*Sqrt[c]*Sqrt[c*x])/(a^(1/4)*b^(1 
/4)*Sqrt[c]*Sqrt[c*x]))]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*Sqrt[ 
a + b*x^2]*(a^(1/4)*Sqrt[c] + b^(1/4)*Sqrt[c*x])) - (b^(1/4)*(c*x)^(3/4)*S 
qrt[-((a*c^2 + b*c^2*x^2)/(Sqrt[a]*Sqrt[b]*c^2*x))]*Sqrt[-((a^(1/4)*Sqrt[c 
] - b^(1/4)*Sqrt[c*x])^2/(a^(1/4)*b^(1/4)*Sqrt[c]*Sqrt[c*x]))]*EllipticF[A 
rcSin[Sqrt[(Sqrt[2]*Sqrt[a]*c + Sqrt[2]*Sqrt[b]*c*x + 2*a^(1/4)*b^(1/4)*Sq 
rt[c]*Sqrt[c*x])/(a^(1/4)*b^(1/4)*Sqrt[c]*Sqrt[c*x])]/2], -2*(1 - Sqrt[2]) 
])/(2*Sqrt[2 + Sqrt[2]]*Sqrt[a + b*x^2]*(a^(1/4)*Sqrt[c] - b^(1/4)*Sqrt[c* 
x]))))/(5*b)))/13
 

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 248

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

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 767

Int[1/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[1/2   Int[(1 - Rt[b/a, 4 
]*x^2)/Sqrt[a + b*x^8], x], x] + Simp[1/2   Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a 
 + b*x^8], x], x] /; FreeQ[{a, b}, x]
 

rule 2422

Int[((c_) + (d_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[(-c) 
*d*x^3*Sqrt[-(c - d*x^2)^2/(c*d*x^2)]*(Sqrt[(-d^2)*((a + b*x^8)/(b*c^2*x^4) 
)]/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]))*EllipticF[ArcSin[(1/2)* 
Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d*x^2)]], -2*(1 - Sqrt[ 
2])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^4 - a*d^4, 0]
 

Maple [F]

\[\int \left (c x \right )^{\frac {5}{4}} \sqrt {b \,x^{2}+a}d x\]

Input:

int((c*x)^(5/4)*(b*x^2+a)^(1/2),x)
 

Output:

int((c*x)^(5/4)*(b*x^2+a)^(1/2),x)
 

Fricas [F]

\[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \left (c x\right )^{\frac {5}{4}} \,d x } \] Input:

integrate((c*x)^(5/4)*(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.94 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.09 \[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\frac {\sqrt {a} c^{\frac {5}{4}} x^{\frac {9}{4}} \Gamma \left (\frac {9}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{8} \\ \frac {17}{8} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{8}\right )} \] Input:

integrate((c*x)**(5/4)*(b*x**2+a)**(1/2),x)
 

Output:

sqrt(a)*c**(5/4)*x**(9/4)*gamma(9/8)*hyper((-1/2, 9/8), (17/8,), b*x**2*ex 
p_polar(I*pi)/a)/(2*gamma(17/8))
 

Maxima [F]

\[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \left (c x\right )^{\frac {5}{4}} \,d x } \] Input:

integrate((c*x)^(5/4)*(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*(c*x)^(5/4), x)
 

Giac [F]

\[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \left (c x\right )^{\frac {5}{4}} \,d x } \] Input:

integrate((c*x)^(5/4)*(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*(c*x)^(5/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\int {\left (c\,x\right )}^{5/4}\,\sqrt {b\,x^2+a} \,d x \] Input:

int((c*x)^(5/4)*(a + b*x^2)^(1/2),x)
 

Output:

int((c*x)^(5/4)*(a + b*x^2)^(1/2), x)
 

Reduce [F]

\[ \int (c x)^{5/4} \sqrt {a+b x^2} \, dx=\frac {4 c^{\frac {5}{4}} \left (4 x^{\frac {1}{4}} \sqrt {b \,x^{2}+a}\, a +5 x^{\frac {9}{4}} \sqrt {b \,x^{2}+a}\, b -\left (\int \frac {\sqrt {b \,x^{2}+a}}{x^{\frac {3}{4}} a +x^{\frac {11}{4}} b}d x \right ) a^{2}\right )}{65 b} \] Input:

int((c*x)^(5/4)*(b*x^2+a)^(1/2),x)
 

Output:

(4*c**(1/4)*c*(4*x**(1/4)*sqrt(a + b*x**2)*a + 5*x**(1/4)*sqrt(a + b*x**2) 
*b*x**2 - int(sqrt(a + b*x**2)/(x**(3/4)*a + x**(3/4)*b*x**2),x)*a**2))/(6 
5*b)