3.3.4 \(\int (f x)^{3/2} (d+e x^2) \sqrt {a+b x^2+c x^4} \, dx\) [204]

3.3.4.1 Optimal result

Integrand size = 31, antiderivative size = 297 \[ \int (f x)^{3/2} \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4} \, dx=\frac {2 d (f x)^{5/2} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{2},-\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 f \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}+\frac {2 e (f x)^{9/2} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {9}{4},-\frac {1}{2},-\frac {1}{2},\frac {13}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{9 f^3 \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \]

output

2/5*d*(f*x)^(5/2)*AppellF1(5/4,-1/2,-1/2,9/4,-2*c*x^2/(b-(-4*a*c+b^2)^(1/2 
)),-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))*(c*x^4+b*x^2+a)^(1/2)/f/(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)+2/9*e* 
(f*x)^(9/2)*AppellF1(9/4,-1/2,-1/2,13/4,-2*c*x^2/(b-(-4*a*c+b^2)^(1/2)),-2 
*c*x^2/(b+(-4*a*c+b^2)^(1/2)))*(c*x^4+b*x^2+a)^(1/2)/f^3/(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)
 

3.3.4.2 Mathematica [A] (verified)

Time = 11.76 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.45 \[ \int (f x)^{3/2} \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4} \, dx=\frac {2 f \sqrt {f x} \left (5 \left (a+b x^2+c x^4\right ) \left (-14 b^2 e+2 b c \left (13 d+5 e x^2\right )+c \left (36 a e+65 c d x^2+45 c e x^4\right )\right )+10 a \left (-13 b c d+7 b^2 e-18 a c e\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},\frac {1}{2},\frac {5}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+2 \left (-39 b^2 c d+130 a c^2 d+21 b^3 e-79 a b c e\right ) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{2925 c^2 \sqrt {a+b x^2+c x^4}} \]

input

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

output

(2*f*Sqrt[f*x]*(5*(a + b*x^2 + c*x^4)*(-14*b^2*e + 2*b*c*(13*d + 5*e*x^2) 
+ c*(36*a*e + 65*c*d*x^2 + 45*c*e*x^4)) + 10*a*(-13*b*c*d + 7*b^2*e - 18*a 
*c*e)*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 + 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])] + 2*(-39*b^2*c*d + 130*a*c^2*d + 21*b^3*e - 79*a*b*c*e)*x^2*S 
qrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + S 
qrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[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])]))/(2925*c^2*Sqrt[a + b*x^2 + c*x^4])
 

3.3.4.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1674, 2009}

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.

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

\(\Big \downarrow \) 1674

\(\displaystyle \int \left (d (f x)^{3/2} \sqrt {a+b x^2+c x^4}+\frac {e (f x)^{7/2} \sqrt {a+b x^2+c x^4}}{f^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d (f x)^{5/2} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{2},-\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 f \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}+\frac {2 e (f x)^{9/2} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {9}{4},-\frac {1}{2},-\frac {1}{2},\frac {13}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{9 f^3 \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\)

input

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

output

(2*d*(f*x)^(5/2)*Sqrt[a + b*x^2 + c*x^4]*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])])/(5* 
f*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])]) + (2*e*(f*x)^(9/2)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[9/4, 
-1/2, -1/2, 13/4, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt 
[b^2 - 4*a*c])])/(9*f^3*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])])
 

3.3.4.3.1 Defintions of rubi rules used

rule 1674

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && N 
eQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.3.4.4 Maple [F]

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

input

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

output

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

3.3.4.5 Fricas [F]

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

input

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

output

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

3.3.4.6 Sympy [F]

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

input

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

output

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

3.3.4.7 Maxima [F]

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

input

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

output

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

3.3.4.8 Giac [F]

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

input

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

output

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

3.3.4.9 Mupad [F(-1)]

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

input

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

output

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