\(\int \frac {x^2 (c+d x)^{3/2}}{(a-b x^2)^2} \, dx\) [648]

Optimal result

Integrand size = 23, antiderivative size = 222 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {2 d \sqrt {c+d x}}{b^2}+\frac {(a d+b c x) \sqrt {c+d x}}{2 b^2 \left (a-b x^2\right )}+\frac {\left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \sqrt {\sqrt {b} c-\sqrt {a} d} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 \sqrt {a} b^{9/4}}-\frac {\sqrt {\sqrt {b} c+\sqrt {a} d} \left (2 \sqrt {b} c+5 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 \sqrt {a} b^{9/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {b} \sqrt {c+d x} (5 a d+b x (c-4 d x))}{-a+b x^2}+\frac {\left (2 \sqrt {b} c+5 \sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a}}-\frac {\left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a}}}{4 b^{5/2}} \] Input:

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

Output:

((-2*Sqrt[b]*Sqrt[c + d*x]*(5*a*d + b*x*(c - 4*d*x)))/(-a + b*x^2) + ((2*S 
qrt[b]*c + 5*Sqrt[a]*d)*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b* 
c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/Sqrt[a] - 
 ((2*Sqrt[b]*c - 5*Sqrt[a]*d)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqr 
t[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/Sqr 
t[a])/(4*b^(5/2))
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.52, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {561, 27, 1672, 27, 2205, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1672

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2205

\(\displaystyle \frac {2 \left (\frac {d^4 \int \left (\frac {4 a \left (b c^2-a d^2\right )}{b}+\frac {5 a \left (b c^2-a d^2\right )^2-7 a b c \left (b c^2-a d^2\right ) (c+d x)}{b d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{4 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^3}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(-1/4*(d^2*Sqrt[c + d*x]*(b*c^2 - a*d^2 - b*c*(c + d*x)))/(b^2*(a - (b* 
c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) + (d^4*((4*a*(b*c 
^2 - a*d^2)*Sqrt[c + d*x])/b + (Sqrt[a]*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)*(2*b 
*c^2 - 3*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sq 
rt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(5/4)*d) - (Sqrt[a]*(Sqrt[b]*c + Sqrt[a]* 
d)^(3/2)*(2*b*c^2 + 3*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*ArcTanh[(b^(1/4)*Sqrt 
[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(5/4)*d)))/(4*a*b*(b*c^2 - a 
*d^2))))/d^3
 

Defintions of rubi rules used

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1672

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q 
, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a 
*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), 
 x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c*x^4)^(p + 1)* 
Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + 
 c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]
 

rule 2009

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

rule 2205

Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte 
grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 
2] && Expon[Px, x^2] > 1
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {2 d \sqrt {d x +c}}{b^{2}}+\frac {2 d \left (\frac {-\frac {b c \left (d x +c \right )^{\frac {3}{2}}}{4}+\left (-\frac {a \,d^{2}}{4}+\frac {b \,c^{2}}{4}\right ) \sqrt {d x +c}}{b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}}+\frac {b \left (-\frac {\left (5 a \,d^{2}+2 b \,c^{2}+7 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-5 a \,d^{2}-2 b \,c^{2}+7 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}\right )}{b^{2}}\)	\(246\)
derivativedivides	\(2 d \left (\frac {\sqrt {d x +c}}{b^{2}}-\frac {\frac {-\frac {b c \left (d x +c \right )^{\frac {3}{2}}}{4}+\left (-\frac {a \,d^{2}}{4}+\frac {b \,c^{2}}{4}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-5 a \,d^{2}-2 b \,c^{2}-7 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (5 a \,d^{2}+2 b \,c^{2}-7 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{b^{2}}\right )\)	\(247\)
default	\(2 d \left (\frac {\sqrt {d x +c}}{b^{2}}-\frac {\frac {-\frac {b c \left (d x +c \right )^{\frac {3}{2}}}{4}+\left (-\frac {a \,d^{2}}{4}+\frac {b \,c^{2}}{4}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-5 a \,d^{2}-2 b \,c^{2}-7 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (5 a \,d^{2}+2 b \,c^{2}-7 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{b^{2}}\right )\)	\(247\)
pseudoelliptic	\(-\frac {5 \left (d \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, b \left (-b \,x^{2}+a \right ) \left (a \,d^{2}+\frac {2 b \,c^{2}}{5}-\frac {7 \sqrt {a b \,d^{2}}\, c}{5}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d \left (a \,d^{2}+\frac {2 b \,c^{2}}{5}+\frac {7 \sqrt {a b \,d^{2}}\, c}{5}\right ) b \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \left (\frac {x \left (-4 d x +c \right ) b}{5}+a d \right ) \sqrt {a b \,d^{2}}\right )\right )}{4 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, b^{2} \left (-b \,x^{2}+a \right )}\)	\(261\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1089 vs. \(2 (165) = 330\).

Time = 0.11 (sec) , antiderivative size = 1089, normalized size of antiderivative = 4.91 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*((b^3*x^2 - a*b^2)*sqrt((a*b^4*sqrt((576*b^2*c^4*d^2 + 1200*a*b*c^2*d^ 
4 + 625*a^2*d^6)/(a*b^9)) + 4*b*c^3 + 45*a*c*d^2)/(a*b^4))*log(-(96*b^2*c^ 
4*d - 500*a*b*c^2*d^3 - 625*a^2*d^5)*sqrt(d*x + c) + (2*a*b^7*c*sqrt((576* 
b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 625*a^2*d^6)/(a*b^9)) - 120*a*b^3*c^2*d^2 
 - 125*a^2*b^2*d^4)*sqrt((a*b^4*sqrt((576*b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 
 625*a^2*d^6)/(a*b^9)) + 4*b*c^3 + 45*a*c*d^2)/(a*b^4))) - (b^3*x^2 - a*b^ 
2)*sqrt((a*b^4*sqrt((576*b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 625*a^2*d^6)/(a* 
b^9)) + 4*b*c^3 + 45*a*c*d^2)/(a*b^4))*log(-(96*b^2*c^4*d - 500*a*b*c^2*d^ 
3 - 625*a^2*d^5)*sqrt(d*x + c) - (2*a*b^7*c*sqrt((576*b^2*c^4*d^2 + 1200*a 
*b*c^2*d^4 + 625*a^2*d^6)/(a*b^9)) - 120*a*b^3*c^2*d^2 - 125*a^2*b^2*d^4)* 
sqrt((a*b^4*sqrt((576*b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 625*a^2*d^6)/(a*b^9 
)) + 4*b*c^3 + 45*a*c*d^2)/(a*b^4))) - (b^3*x^2 - a*b^2)*sqrt(-(a*b^4*sqrt 
((576*b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 625*a^2*d^6)/(a*b^9)) - 4*b*c^3 - 4 
5*a*c*d^2)/(a*b^4))*log(-(96*b^2*c^4*d - 500*a*b*c^2*d^3 - 625*a^2*d^5)*sq 
rt(d*x + c) + (2*a*b^7*c*sqrt((576*b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 625*a^ 
2*d^6)/(a*b^9)) + 120*a*b^3*c^2*d^2 + 125*a^2*b^2*d^4)*sqrt(-(a*b^4*sqrt(( 
576*b^2*c^4*d^2 + 1200*a*b*c^2*d^4 + 625*a^2*d^6)/(a*b^9)) - 4*b*c^3 - 45* 
a*c*d^2)/(a*b^4))) + (b^3*x^2 - a*b^2)*sqrt(-(a*b^4*sqrt((576*b^2*c^4*d^2 
+ 1200*a*b*c^2*d^4 + 625*a^2*d^6)/(a*b^9)) - 4*b*c^3 - 45*a*c*d^2)/(a*b^4) 
)*log(-(96*b^2*c^4*d - 500*a*b*c^2*d^3 - 625*a^2*d^5)*sqrt(d*x + c) - (...
 

Sympy [F(-1)]

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

integrate(x**2*(d*x+c)**(3/2)/(-b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((d*x + c)^(3/2)*x^2/(b*x^2 - a)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (165) = 330\).

Time = 0.21 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.84 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {2 \, \sqrt {d x + c} d}{b^{2}} + \frac {{\left (7 \, a b c d^{3} {\left | b \right |} - 5 \, {\left (\sqrt {a b} b c^{2} d - \sqrt {a b} a d^{3}\right )} {\left | b \right |} {\left | d \right |} - {\left (2 \, b^{2} c^{3} d + 5 \, a b c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{3} c + \sqrt {b^{6} c^{2} - {\left (b^{3} c^{2} - a b^{2} d^{2}\right )} b^{3}}}{b^{3}}}}\right )}{4 \, {\left (a b^{3} d - \sqrt {a b} b^{3} c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (7 \, a b c d^{3} {\left | b \right |} + 5 \, {\left (\sqrt {a b} b c^{2} d - \sqrt {a b} a d^{3}\right )} {\left | b \right |} {\left | d \right |} - {\left (2 \, b^{2} c^{3} d + 5 \, a b c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{3} c - \sqrt {b^{6} c^{2} - {\left (b^{3} c^{2} - a b^{2} d^{2}\right )} b^{3}}}{b^{3}}}}\right )}{4 \, {\left (a b^{3} d + \sqrt {a b} b^{3} c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} - \frac {{\left (d x + c\right )}^{\frac {3}{2}} b c d - \sqrt {d x + c} b c^{2} d + \sqrt {d x + c} a d^{3}}{2 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )} b^{2}} \] Input:

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

Output:

2*sqrt(d*x + c)*d/b^2 + 1/4*(7*a*b*c*d^3*abs(b) - 5*(sqrt(a*b)*b*c^2*d - s 
qrt(a*b)*a*d^3)*abs(b)*abs(d) - (2*b^2*c^3*d + 5*a*b*c*d^3)*abs(b))*arctan 
(sqrt(d*x + c)/sqrt(-(b^3*c + sqrt(b^6*c^2 - (b^3*c^2 - a*b^2*d^2)*b^3))/b 
^3))/((a*b^3*d - sqrt(a*b)*b^3*c)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(d)) + 1 
/4*(7*a*b*c*d^3*abs(b) + 5*(sqrt(a*b)*b*c^2*d - sqrt(a*b)*a*d^3)*abs(b)*ab 
s(d) - (2*b^2*c^3*d + 5*a*b*c*d^3)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^3 
*c - sqrt(b^6*c^2 - (b^3*c^2 - a*b^2*d^2)*b^3))/b^3))/((a*b^3*d + sqrt(a*b 
)*b^3*c)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(d)) - 1/2*((d*x + c)^(3/2)*b*c*d 
 - sqrt(d*x + c)*b*c^2*d + sqrt(d*x + c)*a*d^3)/(((d*x + c)^2*b - 2*(d*x + 
 c)*b*c + b*c^2 - a*d^2)*b^2)
 

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 1694, normalized size of antiderivative = 7.63 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int((x^2*(c + d*x)^(3/2))/(a - b*x^2)^2,x)
 

Output:

(2*d*(c + d*x)^(1/2))/b^2 - atan((a^2*d^6*(c + d*x)^(1/2)*((45*c*d^2)/(64* 
b^4) + c^3/(16*a*b^3) - (25*d^3*(a^3*b^9)^(1/2))/(64*a*b^9) - (3*c^2*d*(a^ 
3*b^9)^(1/2))/(8*a^2*b^8))^(1/2)*50i)/(12*c^5*d^3 + (a*c^3*d^5)/(2*b) - (2 
5*a^2*c*d^7)/(2*b^2) + (125*a*d^8*(a^3*b^9)^(1/2))/(4*b^7) - (5*c^2*d^6*(a 
^3*b^9)^(1/2))/(4*b^6) - (30*c^4*d^4*(a^3*b^9)^(1/2))/(a*b^5)) + (a*c^2*d^ 
4*(c + d*x)^(1/2)*((45*c*d^2)/(64*b^4) + c^3/(16*a*b^3) - (25*d^3*(a^3*b^9 
)^(1/2))/(64*a*b^9) - (3*c^2*d*(a^3*b^9)^(1/2))/(8*a^2*b^8))^(1/2)*48i)/(( 
12*c^5*d^3)/b + (a*c^3*d^5)/(2*b^2) - (25*a^2*c*d^7)/(2*b^3) + (125*a*d^8* 
(a^3*b^9)^(1/2))/(4*b^8) - (5*c^2*d^6*(a^3*b^9)^(1/2))/(4*b^7) - (30*c^4*d 
^4*(a^3*b^9)^(1/2))/(a*b^6)) + (c^3*d^3*(a^3*b^9)^(1/2)*(c + d*x)^(1/2)*(( 
45*c*d^2)/(64*b^4) + c^3/(16*a*b^3) - (25*d^3*(a^3*b^9)^(1/2))/(64*a*b^9) 
- (3*c^2*d*(a^3*b^9)^(1/2))/(8*a^2*b^8))^(1/2)*48i)/(12*a*b^3*c^5*d^3 + (a 
^2*b^2*c^3*d^5)/2 - (25*a^3*b*c*d^7)/2 + (125*a^2*d^8*(a^3*b^9)^(1/2))/(4* 
b^4) - (30*c^4*d^4*(a^3*b^9)^(1/2))/b^2 - (5*a*c^2*d^6*(a^3*b^9)^(1/2))/(4 
*b^3)) + (c*d^5*(a^3*b^9)^(1/2)*(c + d*x)^(1/2)*((45*c*d^2)/(64*b^4) + c^3 
/(16*a*b^3) - (25*d^3*(a^3*b^9)^(1/2))/(64*a*b^9) - (3*c^2*d*(a^3*b^9)^(1/ 
2))/(8*a^2*b^8))^(1/2)*50i)/(12*b^4*c^5*d^3 + (a*b^3*c^3*d^5)/2 - (25*a^2* 
b^2*c*d^7)/2 + (125*a*d^8*(a^3*b^9)^(1/2))/(4*b^3) - (5*c^2*d^6*(a^3*b^9)^ 
(1/2))/(4*b^2) - (30*c^4*d^4*(a^3*b^9)^(1/2))/(a*b)))*((4*a*b^6*c^3 - 25*a 
*d^3*(a^3*b^9)^(1/2) + 45*a^2*b^5*c*d^2 - 24*b*c^2*d*(a^3*b^9)^(1/2))/(...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.69 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {4 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a b c -4 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b^{2} c \,x^{2}-10 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a^{2} d +10 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a b d \,x^{2}+2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a b c -2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c \,x^{2}-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a b c +2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c \,x^{2}+5 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a^{2} d -5 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a b d \,x^{2}-5 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a^{2} d +5 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a b d \,x^{2}+20 \sqrt {d x +c}\, a^{2} b d +4 \sqrt {d x +c}\, a \,b^{2} c x -16 \sqrt {d x +c}\, a \,b^{2} d \,x^{2}}{8 a \,b^{3} \left (-b \,x^{2}+a \right )} \] Input:

int(x^2*(d*x+c)^(3/2)/(-b*x^2+a)^2,x)
 

Output:

(4*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b*c - 4*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - 
b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**2* 
c*x**2 - 10*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/( 
sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*d + 10*sqrt(b)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b 
*c)))*a*b*d*x**2 + 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqr 
t(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b*c - 2*sqrt(a)*sqrt(sqrt 
(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c 
 + d*x))*b**2*c*x**2 - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sq 
rt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b*c + 2*sqrt(a)*sqrt(sqr 
t(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + 
 d*x))*b**2*c*x**2 + 5*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(s 
qrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a**2*d - 5*sqrt(b)*sqrt(s 
qrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqr 
t(c + d*x))*a*b*d*x**2 - 5*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt( 
sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a**2*d + 5*sqrt(b)*sqrt( 
sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt( 
c + d*x))*a*b*d*x**2 + 20*sqrt(c + d*x)*a**2*b*d + 4*sqrt(c + d*x)*a*b**2* 
c*x - 16*sqrt(c + d*x)*a*b**2*d*x**2)/(8*a*b**3*(a - b*x**2))