\(\int \frac {(a+b x^2)^2}{\sqrt {x} (c+d x^2)} \, dx\) [741]

Optimal result

Integrand size = 24, antiderivative size = 200 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}-\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{3/4} d^{9/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx=\frac {4 b \sqrt [4]{d} \sqrt {x} \left (-5 b c+10 a d+b d x^2\right )-\frac {5 \sqrt {2} (b c-a d)^2 \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4}}+\frac {5 \sqrt {2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4}}}{10 d^{9/4}} \] Input:

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

Output:

(4*b*d^(1/4)*Sqrt[x]*(-5*b*c + 10*a*d + b*d*x^2) - (5*Sqrt[2]*(b*c - a*d)^ 
2*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/c^(3/4) 
 + (5*Sqrt[2]*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqr 
t[c] + Sqrt[d]*x)])/c^(3/4))/(10*d^(9/4))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {364, 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.

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

\(\Big \downarrow \) 364

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{3/4} d^{9/4}}-\frac {(b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}-\frac {2 b \sqrt {x} (b c-2 a d)}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 364

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

rule 2009

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {2 \left (b d \,x^{2}+10 a d -5 b c \right ) b \sqrt {x}}{5 d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} c}\)	\(155\)
derivativedivides	\(\frac {2 b \left (\frac {b \,x^{\frac {5}{2}} d}{5}+2 a d \sqrt {x}-b c \sqrt {x}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} c}\)	\(159\)
default	\(\frac {2 b \left (\frac {b \,x^{\frac {5}{2}} d}{5}+2 a d \sqrt {x}-b c \sqrt {x}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} c}\)	\(159\)

Input:

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

Output:

2/5*(b*d*x^2+10*a*d-5*b*c)*b*x^(1/2)/d^2+1/4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d 
^2*(c/d)^(1/4)/c*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 = 0.11 (sec) , antiderivative size = 1102, normalized size of antiderivative = 5.51 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/10*(5*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c 
^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8* 
a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)*log(c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7 
*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5 
*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1 
/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 5*I*d^2*(-(b^8*c^8 - 8*a* 
b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 
 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d 
^9))^(1/4)*log(I*c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 5 
6*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c 
^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4) + (b^2*c^2 - 2*a*b*c*d 
+ a^2*d^2)*sqrt(x)) - 5*I*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6* 
d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^ 
6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)*log(-I*c*d^2*(-( 
b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4 
*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a 
^8*d^8)/(c^3*d^9))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) - 5*d^ 
2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 7 
0*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^ 
7 + a^8*d^8)/(c^3*d^9))^(1/4)*log(-c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 2...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (190) = 380\).

Time = 4.05 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}}{d} & \text {for}\: c = 0 \\\frac {2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}}{c} & \text {for}\: d = 0 \\- \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 c} + \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 c} + \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c} + \frac {4 a b \sqrt {x}}{d} + \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{d} - \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{d} - \frac {2 a b \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d} - \frac {2 b^{2} c \sqrt {x}}{d^{2}} - \frac {b^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {b^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {b^{2} c \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2}} + \frac {2 b^{2} x^{\frac {5}{2}}}{5 d} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((zoo*(-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5), 
 Eq(c, 0) & Eq(d, 0)), ((-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x** 
(5/2)/5)/d, Eq(c, 0)), ((2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(9/ 
2)/9)/c, Eq(d, 0)), (-a**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(2*c 
) + a**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*c) + a**2*(-c/d)**( 
1/4)*atan(sqrt(x)/(-c/d)**(1/4))/c + 4*a*b*sqrt(x)/d + a*b*(-c/d)**(1/4)*l 
og(sqrt(x) - (-c/d)**(1/4))/d - a*b*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1 
/4))/d - 2*a*b*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/d - 2*b**2*c*sqrt 
(x)/d**2 - b**2*c*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(2*d**2) + b* 
*2*c*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*d**2) + b**2*c*(-c/d)** 
(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/d**2 + 2*b**2*x**(5/2)/(5*d), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx=\frac {2 \, {\left (b^{2} d x^{\frac {5}{2}} - 5 \, {\left (b^{2} c - 2 \, a b d\right )} \sqrt {x}\right )}}{5 \, d^{2}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{4 \, d^{2}} \] Input:

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

Output:

2/5*(b^2*d*x^(5/2) - 5*(b^2*c - 2*a*b*d)*sqrt(x))/d^2 + 1/4*(2*sqrt(2)*(b^ 
2*c^2 - 2*a*b*c*d + a^2*d^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(c)*sqrt(sqrt(c)*sqrt(d))) 
 + 2*sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^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(c)*sqrt( 
sqrt(c)*sqrt(d))) + sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(sqrt(2)*c^ 
(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*( 
b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt 
(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/d^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (150) = 300\).

Time = 0.13 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx=\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{3}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{3}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c d^{3}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c d^{3}} + \frac {2 \, {\left (b^{2} d^{4} x^{\frac {5}{2}} - 5 \, b^{2} c d^{3} \sqrt {x} + 10 \, a b d^{4} \sqrt {x}\right )}}{5 \, d^{5}} \] Input:

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

Output:

1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/ 
4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/ 
4))/(c*d^3) + 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d 
 + (c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqr 
t(x))/(c/d)^(1/4))/(c*d^3) + 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3 
)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + 
 x + sqrt(c/d))/(c*d^3) - 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^( 
1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x 
 + sqrt(c/d))/(c*d^3) + 2/5*(b^2*d^4*x^(5/2) - 5*b^2*c*d^3*sqrt(x) + 10*a* 
b*d^4*sqrt(x))/d^5
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1107, normalized size of antiderivative = 5.54 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*b^2*x^(5/2))/(5*d) - x^(1/2)*((2*b^2*c)/d^2 - (4*a*b)/d) + (atan(((((8* 
x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c 
*d^3))/d - ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/ 
(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4)) + (((8*x^(1 
/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3 
))/d + ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/(2*( 
-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4)))/((((8*x^(1/2)* 
(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d 
 - ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/(2*(-c)^ 
(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4)*d^(9/4)) - (((8*x^(1/2)*(a^4*d^ 
4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d + ((a* 
d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/(2*(-c)^(3/4)*d 
^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4)*d^(9/4))))*(a*d - b*c)^2*1i)/((-c)^(3/ 
4)*d^(9/4)) + (atan(((((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 
 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d - ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2 
*c^3*d - 32*a*b*c^2*d^2)*1i)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^ 
(3/4)*d^(9/4)) + (((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a 
*b^3*c^3*d - 4*a^3*b*c*d^3))/d + ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3 
*d - 32*a*b*c^2*d^2)*1i)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4 
)*d^(9/4)))/((((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 10*d**(3/4)*c**(1/4)*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**2 + 20*d**(3/4)*c**(1/4)* 
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 - 10*d**(3/4)*c**(1/4)*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 + 
 10*d**(3/4)*c**(1/4)*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**2 - 20*d**(3/4)*c**(1/4)*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*d + 10*d**(3/4)*c**(1/4)*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 - 5* 
d**(3/4)*c**(1/4)*sqrt(2)*log( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt( 
c) + sqrt(d)*x)*a**2*d**2 + 10*d**(3/4)*c**(1/4)*sqrt(2)*log( - sqrt(x)*d* 
*(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*a*b*c*d - 5*d**(3/4)*c**(1/ 
4)*sqrt(2)*log( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x) 
*b**2*c**2 + 5*d**(3/4)*c**(1/4)*sqrt(2)*log(sqrt(x)*d**(1/4)*c**(1/4)*sqr 
t(2) + sqrt(c) + sqrt(d)*x)*a**2*d**2 - 10*d**(3/4)*c**(1/4)*sqrt(2)*log(s 
qrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*a*b*c*d + 5*d**(3/ 
4)*c**(1/4)*sqrt(2)*log(sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt 
(d)*x)*b**2*c**2 + 80*sqrt(x)*a*b*c*d**2 - 40*sqrt(x)*b**2*c**2*d + 8*sqrt 
(x)*b**2*c*d**2*x**2)/(20*c*d**3)