\(\int \frac {(c+d x^2)^3}{\sqrt {a+b x^2} (e+f x^2)} \, dx\) [328]

Optimal result

Integrand size = 30, antiderivative size = 217 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=-\frac {d^2 (4 b d e-12 b c f+3 a d f) x \sqrt {a+b x^2}}{8 b^2 f^2}+\frac {d^3 x^3 \sqrt {a+b x^2}}{4 b f}+\frac {d \left (3 a^2 d^2 f^2+4 a b d f (d e-3 c f)+8 b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2} f^3}-\frac {(d e-c f)^3 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^3 \sqrt {b e-a f}} \] Output:

-1/8*d^2*(3*a*d*f-12*b*c*f+4*b*d*e)*x*(b*x^2+a)^(1/2)/b^2/f^2+1/4*d^3*x^3* 
(b*x^2+a)^(1/2)/b/f+1/8*d*(3*a^2*d^2*f^2+4*a*b*d*f*(-3*c*f+d*e)+8*b^2*(3*c 
^2*f^2-3*c*d*e*f+d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)/f^3- 
(-c*f+d*e)^3*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/f 
^3/(-a*f+b*e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {\frac {d^2 f x \sqrt {a+b x^2} \left (-3 a d f+2 b \left (-2 d e+6 c f+d f x^2\right )\right )}{b^2}+\frac {8 (d e-c f)^3 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} \sqrt {-b e+a f}}-\frac {d \left (3 a^2 d^2 f^2+4 a b d f (d e-3 c f)+8 b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}}}{8 f^3} \] Input:

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

Output:

((d^2*f*x*Sqrt[a + b*x^2]*(-3*a*d*f + 2*b*(-2*d*e + 6*c*f + d*f*x^2)))/b^2 
 + (8*(d*e - c*f)^3*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/ 
(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*Sqrt[-(b*e) + a*f]) - (d*(3*a^2*d^ 
2*f^2 + 4*a*b*d*f*(d*e - 3*c*f) + 8*b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2)) 
*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(5/2))/(8*f^3)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.40, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {420, 318, 299, 224, 219, 420, 299, 224, 219, 398, 224, 219, 291, 221}

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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {d \left (\frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\)

\(\Big \downarrow \) 420

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\)

Input:

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

Output:

(d*((d*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*b) + ((3*d*(2*b*c - a*d)*x*Sqrt[a 
 + b*x^2])/(2*b) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x 
)/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(4*b)))/f - ((d*e - c*f)*((d*((d*x*Sqrt[a 
 + b*x^2])/(2*b) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2 
*b^(3/2))))/f - ((d*e - c*f)*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sq 
rt[b]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^ 
2])])/(Sqrt[e]*f*Sqrt[b*e - a*f])))/f))/f
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 299

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 420

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[d/b   Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], 
x] + Simp[(b*c - a*d)/b   Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 
)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
 

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(\frac {-b^{\frac {9}{2}} \left (c f -d e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {3 \left (\frac {\left (\frac {8 b^{2} d^{2} e^{2}}{3}+\frac {4 b d f \left (a d -6 b c \right ) e}{3}+f^{2} \left (a^{2} d^{2}-4 a b c d +8 b^{2} c^{2}\right )\right ) b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{4}+d \,b^{\frac {5}{2}} \left (-\frac {b d e}{3}-\frac {\left (\left (-\frac {2 x^{2} d}{3}-4 c \right ) b +a d \right ) f}{4}\right ) \sqrt {b \,x^{2}+a}\, x f \right ) d \sqrt {\left (a f -b e \right ) e}}{2}}{\sqrt {\left (a f -b e \right ) e}\, b^{\frac {9}{2}} f^{3}}\)	\(194\)
risch	\(-\frac {d^{2} x \left (-2 b d f \,x^{2}+3 a d f -12 b c f +4 b d e \right ) \sqrt {b \,x^{2}+a}}{8 b^{2} f^{2}}+\frac {\frac {d \left (3 a^{2} d^{2} f^{2}-12 a b c d \,f^{2}+4 a b \,d^{2} e f +24 b^{2} c^{2} f^{2}-24 b^{2} c d e f +8 b^{2} d^{2} e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}-\frac {4 b^{2} \left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}+\frac {4 b^{2} \left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}}{8 b^{2} f^{2}}\)	\(523\)
default	\(\frac {d \left (\frac {d^{2} e^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+d f \left (3 c f -d e \right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+d^{2} f^{2} \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+\frac {3 c^{2} f^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {3 c d e f \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\right )}{f^{3}}+\frac {\left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{2 f^{3} \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}-\frac {\left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{2 f^{3} \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}\)	\(581\)

Input:

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

Output:

3/2*(-2/3*b^(9/2)*(c*f-d*e)^3*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/ 
2))+(1/4*(8/3*b^2*d^2*e^2+4/3*b*d*f*(a*d-6*b*c)*e+f^2*(a^2*d^2-4*a*b*c*d+8 
*b^2*c^2))*b^2*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+d*b^(5/2)*(-1/3*b*d*e-1/ 
4*((-2/3*x^2*d-4*c)*b+a*d)*f)*(b*x^2+a)^(1/2)*x*f)*d*((a*f-b*e)*e)^(1/2))/ 
((a*f-b*e)*e)^(1/2)/b^(9/2)/f^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (191) = 382\).

Time = 6.02 (sec) , antiderivative size = 1729, normalized size of antiderivative = 7.97 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*((8*b^3*d^3*e^4 - 4*(6*b^3*c*d^2 + a*b^2*d^3)*e^3*f + (24*b^3*c^2*d 
+ 12*a*b^2*c*d^2 - a^2*b*d^3)*e^2*f^2 - 3*(8*a*b^2*c^2*d - 4*a^2*b*c*d^2 + 
 a^3*d^3)*e*f^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 
 4*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*sqr 
t(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2* 
(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a 
*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) + 2*(2*(b^3*d^3*e^2*f^ 
2 - a*b^2*d^3*e*f^3)*x^3 - (4*b^3*d^3*e^3*f - (12*b^3*c*d^2 + a*b^2*d^3)*e 
^2*f^2 + 3*(4*a*b^2*c*d^2 - a^2*b*d^3)*e*f^3)*x)*sqrt(b*x^2 + a))/(b^4*e^2 
*f^3 - a*b^3*e*f^4), -1/8*((8*b^3*d^3*e^4 - 4*(6*b^3*c*d^2 + a*b^2*d^3)*e^ 
3*f + (24*b^3*c^2*d + 12*a*b^2*c*d^2 - a^2*b*d^3)*e^2*f^2 - 3*(8*a*b^2*c^2 
*d - 4*a^2*b*c*d^2 + a^3*d^3)*e*f^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 
 + a)) + 2*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3* 
f^3)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2* 
e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b 
*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - (2*(b^3*d^3* 
e^2*f^2 - a*b^2*d^3*e*f^3)*x^3 - (4*b^3*d^3*e^3*f - (12*b^3*c*d^2 + a*b^2* 
d^3)*e^2*f^2 + 3*(4*a*b^2*c*d^2 - a^2*b*d^3)*e*f^3)*x)*sqrt(b*x^2 + a))/(b 
^4*e^2*f^3 - a*b^3*e*f^4), 1/16*(8*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^ 
3*c^2*d*e*f^2 - b^3*c^3*f^3)*sqrt(-b*e^2 + a*e*f)*arctan(1/2*sqrt(-b*e^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 984, normalized size of antiderivative = 4.53 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x 
**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*c**3*f**3 + 24*sqrt(e)*s 
qrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)* 
sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*c**2*d*e*f**2 - 24*sqrt(e)*sqrt(a*f - b 
*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/ 
(sqrt(e)*sqrt(b)))*b**3*c*d**2*e**2*f + 8*sqrt(e)*sqrt(a*f - b*e)*atan((sq 
rt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqr 
t(b)))*b**3*d**3*e**3 - 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + 
sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*c**3 
*f**3 + 24*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a 
+ b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*c**2*d*e*f**2 - 24* 
sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + 
 sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*c*d**2*e**2*f + 8*sqrt(e)*sqrt 
(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqr 
t(b)*x)/(sqrt(e)*sqrt(b)))*b**3*d**3*e**3 - 3*sqrt(a + b*x**2)*a**2*b*d**3 
*e*f**3*x + 12*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f**3*x - sqrt(a + b*x**2)* 
a*b**2*d**3*e**2*f**2*x + 2*sqrt(a + b*x**2)*a*b**2*d**3*e*f**3*x**3 - 12* 
sqrt(a + b*x**2)*b**3*c*d**2*e**2*f**2*x + 4*sqrt(a + b*x**2)*b**3*d**3*e* 
*3*f*x - 2*sqrt(a + b*x**2)*b**3*d**3*e**2*f**2*x**3 + 3*sqrt(b)*log((sqrt 
(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*d**3*e*f**3 - 12*sqrt(b)*log((s...