3.30.51 \(\int x^4 \sqrt {a+b (c x^2)^{3/2}} \, dx\) [2951]

3.30.51.1 Optimal result

Integrand size = 21, antiderivative size = 709 \[ \int x^4 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {6 a c x^7 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b \left (c x^2\right )^{5/2}}-\frac {24 a^2 x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b^{5/3} \left (c x^2\right )^{5/2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}-\frac {8 \sqrt {2} 3^{3/4} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right ),-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}} \]

output

2/13*x^5*(a+b*(c*x^2)^(3/2))^(1/2)+6/91*a*c*x^7*(a+b*(c*x^2)^(3/2))^(1/2)/ 
b/(c*x^2)^(5/2)-24/91*a^2*x^5*(a+b*(c*x^2)^(3/2))^(1/2)/b^(5/3)/(c*x^2)^(5 
/2)/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^2)^(1/2))-8/91*3^(3/4)*a^(7/3)*x^5*E 
llipticF((a^(1/3)*(1-3^(1/2))+b^(1/3)*(c*x^2)^(1/2))/(a^(1/3)*(1+3^(1/2))+ 
b^(1/3)*(c*x^2)^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*(a^(1/3)+b^(1/3)*(c*x^2)^(1/ 
2))*((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*(c*x^2)^(1/2))/(a^(1/3)*(1+3^( 
1/2))+b^(1/3)*(c*x^2)^(1/2))^2)^(1/2)/b^(5/3)/(c*x^2)^(5/2)/(a+b*(c*x^2)^( 
3/2))^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*(c*x^2)^(1/2))/(a^(1/3)*(1+3^(1/2))+ 
b^(1/3)*(c*x^2)^(1/2))^2)^(1/2)+12/91*3^(1/4)*a^(7/3)*x^5*EllipticE((a^(1/ 
3)*(1-3^(1/2))+b^(1/3)*(c*x^2)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^2) 
^(1/2)),I*3^(1/2)+2*I)*(a^(1/3)+b^(1/3)*(c*x^2)^(1/2))*(1/2*6^(1/2)-1/2*2^ 
(1/2))*((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*(c*x^2)^(1/2))/(a^(1/3)*(1+ 
3^(1/2))+b^(1/3)*(c*x^2)^(1/2))^2)^(1/2)/b^(5/3)/(c*x^2)^(5/2)/(a+b*(c*x^2 
)^(3/2))^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*(c*x^2)^(1/2))/(a^(1/3)*(1+3^(1/2 
))+b^(1/3)*(c*x^2)^(1/2))^2)^(1/2)
 

3.30.51.2 Mathematica [C] (verified)

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

Time = 2.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.15 \[ \int x^4 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {2 x^5 \sqrt {a+b \left (c x^2\right )^{3/2}} \left (a \left (\frac {a+b \left (c x^2\right )^{3/2}}{a}\right )^{3/2}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )\right )}{13 b \left (c x^2\right )^{3/2} \sqrt {\frac {a+b \left (c x^2\right )^{3/2}}{a}}} \]

input

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

output

(2*x^5*Sqrt[a + b*(c*x^2)^(3/2)]*(a*((a + b*(c*x^2)^(3/2))/a)^(3/2) - a*Hy 
pergeometric2F1[-1/2, 2/3, 5/3, -((b*(c*x^2)^(3/2))/a)]))/(13*b*(c*x^2)^(3 
/2)*Sqrt[(a + b*(c*x^2)^(3/2))/a])
 

3.30.51.3 Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {892, 811, 843, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 892

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

\(\Big \downarrow \) 811

\(\displaystyle \frac {x^5 \left (\frac {3}{13} a \int \frac {c^2 x^4}{\sqrt {b \left (c x^2\right )^{3/2}+a}}d\sqrt {c x^2}+\frac {2}{13} \left (c x^2\right )^{5/2} \sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{\left (c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {x^5 \left (\frac {3}{13} a \left (\frac {2 c x^2 \sqrt {a+b \left (c x^2\right )^{3/2}}}{7 b}-\frac {4 a \int \frac {\sqrt {c x^2}}{\sqrt {b \left (c x^2\right )^{3/2}+a}}d\sqrt {c x^2}}{7 b}\right )+\frac {2}{13} \left (c x^2\right )^{5/2} \sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{\left (c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {x^5 \left (\frac {3}{13} a \left (\frac {2 c x^2 \sqrt {a+b \left (c x^2\right )^{3/2}}}{7 b}-\frac {4 a \left (\frac {\int \frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b \left (c x^2\right )^{3/2}+a}}d\sqrt {c x^2}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b \left (c x^2\right )^{3/2}+a}}d\sqrt {c x^2}}{\sqrt [3]{b}}\right )}{7 b}\right )+\frac {2}{13} \left (c x^2\right )^{5/2} \sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{\left (c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {x^5 \left (\frac {3}{13} a \left (\frac {2 c x^2 \sqrt {a+b \left (c x^2\right )^{3/2}}}{7 b}-\frac {4 a \left (\frac {\int \frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b \left (c x^2\right )^{3/2}+a}}d\sqrt {c x^2}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}\right )}{7 b}\right )+\frac {2}{13} \left (c x^2\right )^{5/2} \sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{\left (c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {x^5 \left (\frac {3}{13} a \left (\frac {2 c x^2 \sqrt {a+b \left (c x^2\right )^{3/2}}}{7 b}-\frac {4 a \left (\frac {\frac {2 \sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}\right )}{7 b}\right )+\frac {2}{13} \left (c x^2\right )^{5/2} \sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{\left (c x^2\right )^{5/2}}\)

input

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

output

(x^5*((2*(c*x^2)^(5/2)*Sqrt[a + b*(c*x^2)^(3/2)])/13 + (3*a*((2*c*x^2*Sqrt 
[a + b*(c*x^2)^(3/2)])/(7*b) - (4*a*(((2*Sqrt[a + b*(c*x^2)^(3/2)])/(b^(1/ 
3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])) - (3^(1/4)*Sqrt[2 - Sqrt 
[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 
 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^ 
2])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 
+ Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt 
[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3 
)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)]))/b^(1/3) - (2*(1 - Sqrt[3])*S 
qrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b 
^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/ 
3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[ 
c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/( 
3^(1/4)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[ 
3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)])))/(7*b))) 
/13))/(c*x^2)^(5/2)
 

3.30.51.3.1 Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 811

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

rule 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 843

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

rule 892

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1))   Subst[Int[x^m*(a + b 
*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x 
] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

3.30.51.4 Maple [F]

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

input

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

output

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

3.30.51.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.16 \[ \int x^4 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (12 \, \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} a^{2} {\rm weierstrassZeta}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right )\right ) + {\left (7 \, b^{2} c^{3} x^{5} + 3 \, \sqrt {c x^{2}} a b c x\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}\right )}}{91 \, b^{2} c^{3}} \]

input

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

output

2/91*(12*sqrt(sqrt(c*x^2)*b*c/x)*a^2*weierstrassZeta(0, -4*sqrt(c*x^2)*a/( 
b*c^2*x), weierstrassPInverse(0, -4*sqrt(c*x^2)*a/(b*c^2*x), x)) + (7*b^2* 
c^3*x^5 + 3*sqrt(c*x^2)*a*b*c*x)*sqrt(sqrt(c*x^2)*b*c*x^2 + a))/(b^2*c^3)
 

3.30.51.6 Sympy [F]

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

input

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

output

Integral(x**4*sqrt(a + b*(c*x**2)**(3/2)), x)
 

3.30.51.7 Maxima [F]

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

input

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

output

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

3.30.51.8 Giac [F]

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

input

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

output

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

3.30.51.9 Mupad [F(-1)]

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

input

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

output

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