\(\int (d x)^{3/2} (a+b x^2+c x^4)^{3/2} \, dx\) [1071]

Optimal result

Integrand size = 24, antiderivative size = 148 \[ \int (d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {2 a (d x)^{5/2} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {5}{4},-\frac {3}{2},-\frac {3}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 d \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \] Output:

2/5*a*(d*x)^(5/2)*(c*x^4+b*x^2+a)^(1/2)*AppellF1(5/4,-3/2,-3/2,9/4,-2*c*x^ 
2/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/d/(1+2*c*x^2/(b- 
(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(459\) vs. \(2(148)=296\).

Time = 11.62 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.10 \[ \int (d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {2 d \sqrt {d x} \left (5 \left (-28 b^4 x^2-8 b^3 c x^4+305 b^2 c^2 x^6+480 b c^3 x^8+195 c^4 x^{10}+a^2 c \left (176 b+455 c x^2\right )+a \left (-28 b^3+196 b^2 c x^2+916 b c^2 x^4+650 c^3 x^6\right )\right )+20 a b \left (7 b^2-44 a c\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},\frac {1}{2},\frac {5}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+4 \left (21 b^4-157 a b^2 c+260 a^2 c^2\right ) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{16575 c^2 \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

(2*d*Sqrt[d*x]*(5*(-28*b^4*x^2 - 8*b^3*c*x^4 + 305*b^2*c^2*x^6 + 480*b*c^3 
*x^8 + 195*c^4*x^10 + a^2*c*(176*b + 455*c*x^2) + a*(-28*b^3 + 196*b^2*c*x 
^2 + 916*b*c^2*x^4 + 650*c^3*x^6)) + 20*a*b*(7*b^2 - 44*a*c)*Sqrt[(b - Sqr 
t[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4* 
a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/4, 1/2, 1/2, 5/4, (-2* 
c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 4*(2 
1*b^4 - 157*a*b^2*c + 260*a^2*c^2)*x^2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x 
^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + S 
qrt[b^2 - 4*a*c])]*AppellF1[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 
 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(16575*c^2*Sqrt[a + b*x^2 
+ c*x^4])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1461, 394}

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.

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

\(\Big \downarrow \) 1461

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

\(\Big \downarrow \) 394

\(\displaystyle \frac {2 a (d x)^{5/2} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {5}{4},-\frac {3}{2},-\frac {3}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 d \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\)

Input:

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

Output:

(2*a*(d*x)^(5/2)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[5/4, -3/2, -3/2, 9/4, (- 
2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(5* 
d*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt 
[b^2 - 4*a*c])])
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1461

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^2 + c*x^4)^FracPart[p]/((1 + 2*c*(x^2/(b + R 
t[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^2/(b - Rt[b^2 - 4*a*c, 2])))^F 
racPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^2/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2 
*c*(x^2/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*d*x^5 + b*d*x^3 + a*d*x)*sqrt(c*x^4 + b*x^2 + a)*sqrt(d*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \sqrt {d}\, d \left (208 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} c -20 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{2}+273 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b c \,x^{2}+12 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} x^{2}+171 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c \,x^{4}+117 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{2} x^{6}-520 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{3}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} c^{2}+314 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{3}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{2} c -42 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{3}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{4}-104 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a^{3} c +10 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a^{2} b^{2}\right )}{1989 b c} \] Input:

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

Output:

(2*sqrt(d)*d*(208*sqrt(x)*sqrt(a + b*x**2 + c*x**4)*a**2*c - 20*sqrt(x)*sq 
rt(a + b*x**2 + c*x**4)*a*b**2 + 273*sqrt(x)*sqrt(a + b*x**2 + c*x**4)*a*b 
*c*x**2 + 12*sqrt(x)*sqrt(a + b*x**2 + c*x**4)*b**3*x**2 + 171*sqrt(x)*sqr 
t(a + b*x**2 + c*x**4)*b**2*c*x**4 + 117*sqrt(x)*sqrt(a + b*x**2 + c*x**4) 
*b*c**2*x**6 - 520*int((sqrt(x)*sqrt(a + b*x**2 + c*x**4)*x**3)/(a + b*x** 
2 + c*x**4),x)*a**2*c**2 + 314*int((sqrt(x)*sqrt(a + b*x**2 + c*x**4)*x**3 
)/(a + b*x**2 + c*x**4),x)*a*b**2*c - 42*int((sqrt(x)*sqrt(a + b*x**2 + c* 
x**4)*x**3)/(a + b*x**2 + c*x**4),x)*b**4 - 104*int((sqrt(x)*sqrt(a + b*x* 
*2 + c*x**4))/(a*x + b*x**3 + c*x**5),x)*a**3*c + 10*int((sqrt(x)*sqrt(a + 
 b*x**2 + c*x**4))/(a*x + b*x**3 + c*x**5),x)*a**2*b**2))/(1989*b*c)