\(\int \frac {\sqrt {x} (a+b x^2)^2}{(c+d x^2)^2} \, dx\) [750]

Optimal result

Integrand size = 24, antiderivative size = 241 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {2 b^2 x^{3/2}}{3 d^2}+\frac {(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (7 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}-\frac {(b c-a d) (7 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}}+\frac {(b c-a d) (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} c^{5/4} d^{11/4}} \] Output:

2/3*b^2*x^(3/2)/d^2+1/2*(-a*d+b*c)^2*x^(3/2)/c/d^2/(d*x^2+c)+1/8*(-a*d+b*c 
)*(a*d+7*b*c)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(5/4)/d^ 
(11/4)-1/8*(-a*d+b*c)*(a*d+7*b*c)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4) 
)*2^(1/2)/c^(5/4)/d^(11/4)+1/8*(-a*d+b*c)*(a*d+7*b*c)*arctanh(2^(1/2)*c^(1 
/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(5/4)/d^(11/4)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{c} d^{3/4} x^{3/2} \left (-6 a b c d+3 a^2 d^2+b^2 c \left (7 c+4 d x^2\right )\right )}{c+d x^2}+3 \sqrt {2} \left (7 b^2 c^2-6 a b c d-a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+3 \sqrt {2} \left (7 b^2 c^2-6 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{24 c^{5/4} d^{11/4}} \] Input:

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

Output:

((4*c^(1/4)*d^(3/4)*x^(3/2)*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(7*c + 4*d*x^2 
)))/(c + d*x^2) + 3*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt 
[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + 3*Sqrt[2]*(7*b^2*c^2 
 - 6*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] 
 + Sqrt[d]*x)])/(24*c^(5/4)*d^(11/4))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {366, 27, 25, 363, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 366

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {\sqrt {x} \left (3 b^2 c^2-4 b^2 d x^2 c-6 a b d c-a^2 d^2\right )}{d x^2+c}dx}{4 c d^2}+\frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {\sqrt {x} \left (3 b^2 c^2-4 b^2 d x^2 c-6 a b d c-a^2 d^2\right )}{d x^2+c}dx}{4 c d^2}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(b c-a d) (a d+7 b c) \int \frac {\sqrt {x}}{d x^2+c}dx-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \int \frac {x}{d x^2+c}d\sqrt {x}-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (a d+7 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )-\frac {8}{3} b^2 c x^{3/2}}{4 c d^2}\)

Input:

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

Output:

((b*c - a*d)^2*x^(3/2))/(2*c*d^2*(c + d*x^2)) - ((-8*b^2*c*x^(3/2))/3 + 2* 
(b*c - a*d)*(7*b*c + a*d)*((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4) 
]/(Sqrt[2]*c^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4 
)]/(Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[d]) - (-1/2*Log[Sqrt[c] - Sqrt[2]*c^ 
(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] 
 + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4) 
))/(2*Sqrt[d])))/(4*c*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 363

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

rule 366

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

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}}+\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (x^{2} d +c \right )}+\frac {\left (a d +7 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{d^{2}}\)	\(168\)
derivativedivides	\(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{2 c \left (x^{2} d +c \right )}+\frac {\left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{2}}\)	\(187\)
default	\(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{2 c \left (x^{2} d +c \right )}+\frac {\left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{2}}\)	\(187\)

Input:

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

Output:

2/3*b^2*x^(3/2)/d^2+1/d^2*(2*a*d-2*b*c)*(1/4*(a*d-b*c)/c*x^(3/2)/(d*x^2+c) 
+1/32*(a*d+7*b*c)/c/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/ 
2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1 
/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 1441, normalized size of antiderivative = 5.98 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(3*(c*d^3*x^2 + c^2*d^2)*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^ 
2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3* 
c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^(1/4 
)*log(c^4*d^8*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 
2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^ 
6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^(3/4) - (343*b^6*c^6 
 - 882*a*b^5*c^5*d + 609*a^2*b^4*c^4*d^2 + 36*a^3*b^3*c^3*d^3 - 87*a^4*b^2 
*c^2*d^4 - 18*a^5*b*c*d^5 - a^6*d^6)*sqrt(x)) - 3*(I*c*d^3*x^2 + I*c^2*d^2 
)*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5 
*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^ 
6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^(1/4)*log(I*c^4*d^8*(-(2401*b^8* 
c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 143 
4*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a^7*b*c 
*d^7 + a^8*d^8)/(c^5*d^11))^(3/4) - (343*b^6*c^6 - 882*a*b^5*c^5*d + 609*a 
^2*b^4*c^4*d^2 + 36*a^3*b^3*c^3*d^3 - 87*a^4*b^2*c^2*d^4 - 18*a^5*b*c*d^5 
- a^6*d^6)*sqrt(x)) - 3*(-I*c*d^3*x^2 - I*c^2*d^2)*(-(2401*b^8*c^8 - 8232* 
a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c 
^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8* 
d^8)/(c^5*d^11))^(1/4)*log(-I*c^4*d^8*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 
 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 3...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{2}} - \frac {{\left (7 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, c d^{2}} \] Input:

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

Output:

1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^(3/2)/(c*d^3*x^2 + c^2*d^2) + 2/3*b^ 
2*x^(3/2)/d^2 - 1/16*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*(2*sqrt(2)*arctan(1 
/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt 
(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqr 
t(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqr 
t(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqr 
t(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4) 
*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(c*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (181) = 362\).

Time = 0.13 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{2}} + \frac {b^{2} c^{2} x^{\frac {3}{2}} - 2 \, a b c d x^{\frac {3}{2}} + a^{2} d^{2} x^{\frac {3}{2}}}{2 \, {\left (d x^{2} + c\right )} c d^{2}} - \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{2} d^{5}} - \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{2} d^{5}} + \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{2} d^{5}} - \frac {\sqrt {2} {\left (7 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{2} d^{5}} \] Input:

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

Output:

2/3*b^2*x^(3/2)/d^2 + 1/2*(b^2*c^2*x^(3/2) - 2*a*b*c*d*x^(3/2) + a^2*d^2*x 
^(3/2))/((d*x^2 + c)*c*d^2) - 1/8*sqrt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(c* 
d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c 
/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^5) - 1/8*sqrt(2)*(7*(c*d^3)^(3/ 
4)*b^2*c^2 - 6*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2* 
sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^5) + 1/16*sq 
rt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a 
^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^5) - 1/16* 
sqrt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4) 
*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^5)
 

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {2\,b^2\,x^{3/2}}{3\,d^2}+\frac {x^{3/2}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c\,\left (d^3\,x^2+c\,d^2\right )}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+7\,b\,c\right )}{4\,{\left (-c\right )}^{5/4}\,d^{11/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{5/4}\,d^{11/4}} \] Input:

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

Output:

(2*b^2*x^(3/2))/(3*d^2) + (x^(3/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(2*c*( 
c*d^2 + d^3*x^2)) - (atan((d^(1/4)*x^(1/2))/(-c)^(1/4))*(a*d - b*c)*(a*d + 
 7*b*c))/(4*(-c)^(5/4)*d^(11/4)) - (atan((d^(1/4)*x^(1/2)*1i)/(-c)^(1/4))* 
(a*d - b*c)*(a*d + 7*b*c)*1i)/(4*(-c)^(5/4)*d^(11/4))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 981, normalized size of antiderivative = 4.07 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x 
)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*c*d**2 - 6*d**(1/4)*c**(3/4)* 
sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c** 
(1/4)*sqrt(2)))*a**2*d**3*x**2 - 36*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/ 
4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b* 
c**2*d - 36*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2* 
sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d**2*x**2 + 42*d**(1/4 
)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d 
**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**3 + 42*d**(1/4)*c**(3/4)*sqrt(2)*atan(( 
d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)) 
)*b**2*c**2*d*x**2 + 6*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*s 
qrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*c*d**2 + 6*d 
**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt( 
d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*d**3*x**2 + 36*d**(1/4)*c**(3/4)*sqr 
t(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/ 
4)*sqrt(2)))*a*b*c**2*d + 36*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**( 
1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d**2* 
x**2 - 42*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sq 
rt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**3 - 42*d**(1/4)*c**(3/ 
4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/...