3.3.17 \(\int \frac {\sqrt {f x} (d+e x^2)}{(a+b x^2+c x^4)^{3/2}} \, dx\) [217]

3.3.17.1 Optimal result

Integrand size = 31, antiderivative size = 303 \[ \int \frac {\sqrt {f x} \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 d (f x)^{3/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {3}{2},\frac {3}{2},\frac {7}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 a f \sqrt {a+b x^2+c x^4}}+\frac {2 e (f x)^{7/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {3}{2},\frac {3}{2},\frac {11}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{7 a f^3 \sqrt {a+b x^2+c x^4}} \]

output

2/3*d*(f*x)^(3/2)*AppellF1(3/4,3/2,3/2,7/4,-2*c*x^2/(b-(-4*a*c+b^2)^(1/2)) 
,-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))*(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)/a/f/(c*x^4+b*x^2+a)^(1/2)+2/7*e* 
(f*x)^(7/2)*AppellF1(7/4,3/2,3/2,11/4,-2*c*x^2/(b-(-4*a*c+b^2)^(1/2)),-2*c 
*x^2/(b+(-4*a*c+b^2)^(1/2)))*(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)/a/f^3/(c*x^4+b*x^2+a)^(1/2)
 

3.3.17.2 Mathematica [A] (verified)

Time = 11.60 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {f x} \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \sqrt {f x} \left (-21 b^2 d+21 b \left (a e-c d x^2\right )+42 a c \left (d+e x^2\right )+7 \left (b^2 d+2 a c d-3 a b e\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},\frac {1}{2},\frac {7}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+9 c (b d-2 a e) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},\frac {1}{2},\frac {11}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{21 a \left (-b^2+4 a c\right ) \sqrt {a+b x^2+c x^4}} \]

input

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

output

(x*Sqrt[f*x]*(-21*b^2*d + 21*b*(a*e - c*d*x^2) + 42*a*c*(d + e*x^2) + 7*(b 
^2*d + 2*a*c*d - 3*a*b*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[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2* 
c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 9*c*(b*d - 2*a*e)*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 + Sqrt[b^2 - 4*a*c])]*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x 
^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(21*a*( 
-b^2 + 4*a*c)*Sqrt[a + b*x^2 + c*x^4])
 

3.3.17.3 Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 303, 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.

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

\(\Big \downarrow \) 1674

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d (f x)^{3/2} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {3}{4},\frac {3}{2},\frac {3}{2},\frac {7}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 a f \sqrt {a+b x^2+c x^4}}+\frac {2 e (f x)^{7/2} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {7}{4},\frac {3}{2},\frac {3}{2},\frac {11}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{7 a f^3 \sqrt {a+b x^2+c x^4}}\)

input

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

output

(2*d*(f*x)^(3/2)*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])]*AppellF1[3/4, 3/2, 3/2, 7/4, (-2*c*x^2)/(b 
- Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(3*a*f*Sqrt[a + 
 b*x^2 + c*x^4]) + (2*e*(f*x)^(7/2)*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])]*AppellF1[7/4, 3/2, 3/2, 
11/4, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c 
])])/(7*a*f^3*Sqrt[a + b*x^2 + c*x^4])
 

3.3.17.3.1 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]
 

3.3.17.4 Maple [F]

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

input

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

output

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

3.3.17.5 Fricas [F]

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

input

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

output

integral(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*sqrt(f*x)/(c^2*x^8 + 2*b*c*x^ 
6 + (b^2 + 2*a*c)*x^4 + 2*a*b*x^2 + a^2), x)
 

3.3.17.6 Sympy [F]

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

input

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

output

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

3.3.17.7 Maxima [F]

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

input

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

output

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

3.3.17.8 Giac [F]

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

input

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

output

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

3.3.17.9 Mupad [F(-1)]

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

input

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

output

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