\(\int \sqrt {x} (b x^2+c x^4)^{3/2} \, dx\) [241]

Optimal result

Integrand size = 21, antiderivative size = 203 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^{3/2} \, dx=-\frac {8 b^3 \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}+\frac {8 b^2 x^{3/2} \sqrt {b x^2+c x^4}}{385 c}+\frac {4}{55} b x^{7/2} \sqrt {b x^2+c x^4}+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}+\frac {4 b^{15/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{231 c^{9/4} \sqrt {b x^2+c x^4}} \] Output:

-8/231*b^3*(c*x^4+b*x^2)^(1/2)/c^2/x^(1/2)+8/385*b^2*x^(3/2)*(c*x^4+b*x^2) 
^(1/2)/c+4/55*b*x^(7/2)*(c*x^4+b*x^2)^(1/2)+2/15*x^(3/2)*(c*x^4+b*x^2)^(3/ 
2)+4/231*b^(15/4)*x*(b^(1/2)+c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c^(1/2)*x)^2)^ 
(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x^(1/2)/b^(1/4)),1/2*2^(1/2))/c^(9/ 
4)/(c*x^4+b*x^2)^(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 = 101, normalized size of antiderivative = 0.50 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (-\left (\left (5 b-11 c x^2\right ) \left (b+c x^2\right )^2 \sqrt {1+\frac {c x^2}{b}}\right )+5 b^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{b}\right )\right )}{165 c^2 \sqrt {x} \sqrt {1+\frac {c x^2}{b}}} \] Input:

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

Output:

(2*Sqrt[x^2*(b + c*x^2)]*(-((5*b - 11*c*x^2)*(b + c*x^2)^2*Sqrt[1 + (c*x^2 
)/b]) + 5*b^3*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(165*c^2*S 
qrt[x]*Sqrt[1 + (c*x^2)/b])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1426, 1426, 1429, 1429, 1431, 266, 761}

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.

\(\displaystyle \int \sqrt {x} \left (b x^2+c x^4\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1426

\(\displaystyle \frac {2}{5} b \int x^{5/2} \sqrt {c x^4+b x^2}dx+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

\(\Big \downarrow \) 1426

\(\displaystyle \frac {2}{5} b \left (\frac {2}{11} b \int \frac {x^{9/2}}{\sqrt {c x^4+b x^2}}dx+\frac {2}{11} x^{7/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

\(\Big \downarrow \) 1429

\(\displaystyle \frac {2}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {5 b \int \frac {x^{5/2}}{\sqrt {c x^4+b x^2}}dx}{7 c}\right )+\frac {2}{11} x^{7/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

\(\Big \downarrow \) 1429

\(\displaystyle \frac {2}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {5 b \left (\frac {2 \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {b \int \frac {\sqrt {x}}{\sqrt {c x^4+b x^2}}dx}{3 c}\right )}{7 c}\right )+\frac {2}{11} x^{7/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

\(\Big \downarrow \) 1431

\(\displaystyle \frac {2}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {5 b \left (\frac {2 \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {b x \sqrt {b+c x^2} \int \frac {1}{\sqrt {x} \sqrt {c x^2+b}}dx}{3 c \sqrt {b x^2+c x^4}}\right )}{7 c}\right )+\frac {2}{11} x^{7/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {5 b \left (\frac {2 \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {2 b x \sqrt {b+c x^2} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{3 c \sqrt {b x^2+c x^4}}\right )}{7 c}\right )+\frac {2}{11} x^{7/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {2}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {5 b \left (\frac {2 \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {b^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 c^{5/4} \sqrt {b x^2+c x^4}}\right )}{7 c}\right )+\frac {2}{11} x^{7/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{15} x^{3/2} \left (b x^2+c x^4\right )^{3/2}\)

Input:

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

Output:

(2*x^(3/2)*(b*x^2 + c*x^4)^(3/2))/15 + (2*b*((2*x^(7/2)*Sqrt[b*x^2 + c*x^4 
])/11 + (2*b*((2*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(7*c) - (5*b*((2*Sqrt[b*x^2 
+ c*x^4])/(3*c*Sqrt[x]) - (b^(3/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2 
)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 
1/2])/(3*c^(5/4)*Sqrt[b*x^2 + c*x^4])))/(7*c)))/11))/5
 

Defintions of rubi rules used

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 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 1426

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(d*x)^(m + 1)*((b*x^2 + c*x^4)^p/(d*(m + 4*p + 1))), x] + Simp[2*b*(p/(d^2 
*(m + 4*p + 1)))   Int[(d*x)^(m + 2)*(b*x^2 + c*x^4)^(p - 1), x], x] /; Fre 
eQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0]
 

rule 1429

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[d^3*(d*x)^(m - 3)*((b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[b 
*d^2*((m + 2*p - 1)/(c*(m + 4*p + 1)))   Int[(d*x)^(m - 2)*(b*x^2 + c*x^4)^ 
p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && GtQ[m + 2*p - 1, 
 0] && NeQ[m + 4*p + 1, 0]
 

rule 1431

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(b*x^2 + c*x^4)^p/((d*x)^(2*p)*(b + c*x^2)^p)   Int[(d*x)^(m + 2*p)*(b + c 
*x^2)^p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (77 c^{5} x^{9}+196 c^{4} x^{7} b +10 b^{4} \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+131 b^{2} c^{3} x^{5}-8 c^{2} x^{3} b^{3}-20 x c \,b^{4}\right )}{1155 x^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{2} c^{3}}\)	\(168\)
risch	\(-\frac {2 \left (-77 c^{3} x^{6}-119 b \,c^{2} x^{4}-12 b^{2} c \,x^{2}+20 b^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{1155 c^{2} \sqrt {x}}+\frac {4 b^{4} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{231 c^{3} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\)	\(202\)

Input:

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

Output:

2/1155*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2*(77*c^5*x^9+196*c^4*x^7*b+1 
0*b^4*(-b*c)^(1/2)*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+ 
(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-c/(-b*c)^(1/2)*x)^(1/2)*EllipticF(((c* 
x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))+131*b^2*c^3*x^5-8*c^2*x^3 
*b^3-20*x*c*b^4)/c^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.39 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \, {\left (20 \, b^{4} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (77 \, c^{4} x^{6} + 119 \, b c^{3} x^{4} + 12 \, b^{2} c^{2} x^{2} - 20 \, b^{3} c\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{1155 \, c^{3} x} \] Input:

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

Output:

2/1155*(20*b^4*sqrt(c)*x*weierstrassPInverse(-4*b/c, 0, x) + (77*c^4*x^6 + 
 119*b*c^3*x^4 + 12*b^2*c^2*x^2 - 20*b^3*c)*sqrt(c*x^4 + b*x^2)*sqrt(x))/( 
c^3*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {x} \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {-\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{3}}{231}+\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{2} c \,x^{2}}{385}+\frac {34 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b \,c^{2} x^{4}}{165}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, c^{3} x^{6}}{15}+\frac {4 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{3}+b x}d x \right ) b^{4}}{231}}{c^{2}} \] Input:

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

Output:

(2*( - 20*sqrt(x)*sqrt(b + c*x**2)*b**3 + 12*sqrt(x)*sqrt(b + c*x**2)*b**2 
*c*x**2 + 119*sqrt(x)*sqrt(b + c*x**2)*b*c**2*x**4 + 77*sqrt(x)*sqrt(b + c 
*x**2)*c**3*x**6 + 10*int((sqrt(x)*sqrt(b + c*x**2))/(b*x + c*x**3),x)*b** 
4))/(1155*c**2)