\(\int \frac {x^2 (c+d x)^{3/2}}{(a-b x^2)^3} \, dx\) [699]

Optimal result

Integrand size = 23, antiderivative size = 266 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^3} \, dx=\frac {(a d+b c x) \sqrt {c+d x}}{4 b^2 \left (a-b x^2\right )^2}-\frac {(9 a d+2 b c x) \sqrt {c+d x}}{16 a b^2 \left (a-b x^2\right )}+\frac {\left (4 b c^2-2 \sqrt {a} \sqrt {b} c d-5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{3/2} b^{9/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\left (4 b c^2+2 \sqrt {a} \sqrt {b} c d-5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{3/2} b^{9/4} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

1/4*(b*c*x+a*d)*(d*x+c)^(1/2)/b^2/(-b*x^2+a)^2-1/16*(2*b*c*x+9*a*d)*(d*x+c 
)^(1/2)/a/b^2/(-b*x^2+a)+1/32*(4*b*c^2-2*a^(1/2)*b^(1/2)*c*d-5*a*d^2)*arct 
anh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(3/2)/b^(9/4)/(b^ 
(1/2)*c-a^(1/2)*d)^(1/2)-1/32*(4*b*c^2+2*a^(1/2)*b^(1/2)*c*d-5*a*d^2)*arct 
anh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(3/2)/b^(9/4)/(b^ 
(1/2)*c+a^(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {c+d x} \left (-5 a^2 d+2 b^2 c x^3+a b x (2 c+9 d x)\right )}{\left (a-b x^2\right )^2}-\frac {\left (4 b c^2+2 \sqrt {a} \sqrt {b} c d-5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {\left (4 b c^2-2 \sqrt {a} \sqrt {b} c d-5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{32 a^{3/2} b^2} \] Input:

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

Output:

((2*Sqrt[a]*Sqrt[c + d*x]*(-5*a^2*d + 2*b^2*c*x^3 + a*b*x*(2*c + 9*d*x)))/ 
(a - b*x^2)^2 - ((4*b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*ArcTan[(Sqrt[ 
-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/Sqrt[ 
-(b*c) - Sqrt[a]*Sqrt[b]*d] + ((4*b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2) 
*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt 
[a]*d)])/Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d])/(32*a^(3/2)*b^2)
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.58, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {561, 27, 1672, 27, 2206, 27, 1480, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1672

\(\displaystyle \frac {2 \left (\frac {d^4 \int \frac {2 \left (\frac {a \left (b c^2-a d^2\right )^2}{b d^2}+8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^2-5 a c \left (a-\frac {b c^2}{d^2}\right ) (c+d x)\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{8 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \int \frac {\frac {a \left (b c^2-a d^2\right )^2}{b d^2}+8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^2-5 a c \left (a-\frac {b c^2}{d^2}\right ) (c+d x)}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{8 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {d^4 \int -\frac {2 a \left (b c^2-a d^2\right )^2 \left (2 b c^2+2 b (c+d x) c-5 a d^2\right )}{d^6 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (-9 a d^2+2 b c^2-2 b c (c+d x)\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{8 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (-9 a d^2+2 b c^2-2 b c (c+d x)\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\left (b c^2-a d^2\right ) \int \frac {2 b c^2+2 b (c+d x) c-5 a d^2}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 b d^2}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{8 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (-9 a d^2+2 b c^2-2 b c (c+d x)\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\left (b c^2-a d^2\right ) \left (\frac {\sqrt {b} \left (2 \sqrt {a} \sqrt {b} c d-5 a d^2+4 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b} c d-5 a d^2+4 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )}{4 b d^2}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{8 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (-9 a d^2+2 b c^2-2 b c (c+d x)\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\left (b c^2-a d^2\right ) \left (\frac {d \left (2 \sqrt {a} \sqrt {b} c d-5 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-2 \sqrt {a} \sqrt {b} c d-5 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 b d^2}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (-a d^2+b c^2-b c (c+d x)\right )}{8 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

Input:

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

Output:

(2*(-1/8*(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)^2) + (d^4*(((b*c^2 
 - a*d^2)*Sqrt[c + d*x]*(2*b*c^2 - 9*a*d^2 - 2*b*c*(c + d*x)))/(4*b*d^2*(a 
 - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) - ((b*c^2 - 
 a*d^2)*(-1/2*(d*(4*b*c^2 - 2*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]])/(Sqrt[a]*b^(1/4)*Sqrt[Sqr 
t[b]*c - Sqrt[a]*d]) + (d*(4*b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*ArcT 
anh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^(1/ 
4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/(4*b*d^2)))/(8*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 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 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 1480

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

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 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(-\frac {5 \left (-\frac {d b \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-\frac {4 b \,c^{2}}{5}+\frac {2 \sqrt {a b \,d^{2}}\, c}{5}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+\left (-\frac {d b \left (a \,d^{2}-\frac {4 b \,c^{2}}{5}-\frac {2 \sqrt {a b \,d^{2}}\, c}{5}\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+\sqrt {d x +c}\, \left (-\frac {2 b^{2} c \,x^{3}}{5}-\frac {2 x \left (\frac {9 d x}{2}+c \right ) a b}{5}+a^{2} d \right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{16 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, a \,b^{2} \left (-b \,x^{2}+a \right )^{2}}\)	\(281\)
default	\(2 d^{3} \left (\frac {\frac {c \left (d x +c \right )^{\frac {7}{2}}}{16 a \,d^{2}}+\frac {3 \left (3 a \,d^{2}-2 b \,c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32 a b \,d^{2}}-\frac {c \left (8 a \,d^{2}-3 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16 a b \,d^{2}}-\frac {\left (a \,d^{2}-b \,c^{2}\right ) \left (5 a \,d^{2}-2 b \,c^{2}\right ) \sqrt {d x +c}}{32 a \,b^{2} d^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {-\frac {\left (-5 a \,d^{2}+4 b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (5 a \,d^{2}-4 b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{32 a b \,d^{2}}\right )\)	\(330\)
derivativedivides	\(-2 d^{3} \left (-\frac {\frac {c \left (d x +c \right )^{\frac {7}{2}}}{16 a \,d^{2}}+\frac {3 \left (3 a \,d^{2}-2 b \,c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32 a b \,d^{2}}-\frac {c \left (8 a \,d^{2}-3 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16 a b \,d^{2}}-\frac {\left (a \,d^{2}-b \,c^{2}\right ) \left (5 a \,d^{2}-2 b \,c^{2}\right ) \sqrt {d x +c}}{32 a \,b^{2} d^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {-\frac {\left (-5 a \,d^{2}+4 b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (5 a \,d^{2}-4 b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{32 a b \,d^{2}}\right )\)	\(331\)

Input:

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

Output:

-5/16/(a*b*d^2)^(1/2)*(-1/2*d*b*(-b*x^2+a)^2*(a*d^2-4/5*b*c^2+2/5*(a*b*d^2 
)^(1/2)*c)*((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))+(-1/2*d*b*(a*d^2-4/5*b*c^2-2/5*(a*b*d^2)^(1/2)*c 
)*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+(d 
*x+c)^(1/2)*(-2/5*b^2*c*x^3-2/5*x*(9/2*d*x+c)*a*b+a^2*d)*(a*b*d^2)^(1/2)*( 
(b*c+(a*b*d^2)^(1/2))*b)^(1/2))*((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))/((b*c+(a 
*b*d^2)^(1/2))*b)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/a/b^2/(-b*x^2+a)^ 
2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2197 vs. \(2 (208) = 416\).

Time = 0.17 (sec) , antiderivative size = 2197, normalized size of antiderivative = 8.26 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*((a*b^4*x^4 - 2*a^2*b^3*x^2 + a^3*b^2)*sqrt((16*b^2*c^5 - 52*a*b*c^3* 
d^2 + 45*a^2*c*d^4 + (a^3*b^5*c^2 - a^4*b^4*d^2)*sqrt((256*b^2*c^4*d^6 - 8 
00*a*b*c^2*d^8 + 625*a^2*d^10)/(a^3*b^11*c^4 - 2*a^4*b^10*c^2*d^2 + a^5*b^ 
9*d^4)))/(a^3*b^5*c^2 - a^4*b^4*d^2))*log(-(256*b^3*c^6*d^3 - 1104*a*b^2*c 
^4*d^5 + 1500*a^2*b*c^2*d^7 - 625*a^3*d^9)*sqrt(d*x + c) + (32*a^2*b^4*c^4 
*d^4 - 130*a^3*b^3*c^2*d^6 + 125*a^4*b^2*d^8 + (4*a^3*b^9*c^5 - 11*a^4*b^8 
*c^3*d^2 + 7*a^5*b^7*c*d^4)*sqrt((256*b^2*c^4*d^6 - 800*a*b*c^2*d^8 + 625* 
a^2*d^10)/(a^3*b^11*c^4 - 2*a^4*b^10*c^2*d^2 + a^5*b^9*d^4)))*sqrt((16*b^2 
*c^5 - 52*a*b*c^3*d^2 + 45*a^2*c*d^4 + (a^3*b^5*c^2 - a^4*b^4*d^2)*sqrt((2 
56*b^2*c^4*d^6 - 800*a*b*c^2*d^8 + 625*a^2*d^10)/(a^3*b^11*c^4 - 2*a^4*b^1 
0*c^2*d^2 + a^5*b^9*d^4)))/(a^3*b^5*c^2 - a^4*b^4*d^2))) - (a*b^4*x^4 - 2* 
a^2*b^3*x^2 + a^3*b^2)*sqrt((16*b^2*c^5 - 52*a*b*c^3*d^2 + 45*a^2*c*d^4 + 
(a^3*b^5*c^2 - a^4*b^4*d^2)*sqrt((256*b^2*c^4*d^6 - 800*a*b*c^2*d^8 + 625* 
a^2*d^10)/(a^3*b^11*c^4 - 2*a^4*b^10*c^2*d^2 + a^5*b^9*d^4)))/(a^3*b^5*c^2 
 - a^4*b^4*d^2))*log(-(256*b^3*c^6*d^3 - 1104*a*b^2*c^4*d^5 + 1500*a^2*b*c 
^2*d^7 - 625*a^3*d^9)*sqrt(d*x + c) - (32*a^2*b^4*c^4*d^4 - 130*a^3*b^3*c^ 
2*d^6 + 125*a^4*b^2*d^8 + (4*a^3*b^9*c^5 - 11*a^4*b^8*c^3*d^2 + 7*a^5*b^7* 
c*d^4)*sqrt((256*b^2*c^4*d^6 - 800*a*b*c^2*d^8 + 625*a^2*d^10)/(a^3*b^11*c 
^4 - 2*a^4*b^10*c^2*d^2 + a^5*b^9*d^4)))*sqrt((16*b^2*c^5 - 52*a*b*c^3*d^2 
 + 45*a^2*c*d^4 + (a^3*b^5*c^2 - a^4*b^4*d^2)*sqrt((256*b^2*c^4*d^6 - 8...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (208) = 416\).

Time = 0.25 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.04 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^3} \, dx=-\frac {{\left (2 \, \sqrt {a b} a^{2} c d^{3} {\left | b \right |} + {\left (2 \, a b c^{2} d - 5 \, a^{2} d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (4 \, \sqrt {a b} a b c^{3} d - 5 \, \sqrt {a b} a^{2} c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b^{3} c + \sqrt {a^{2} b^{6} c^{2} - {\left (a b^{3} c^{2} - a^{2} b^{2} d^{2}\right )} a b^{3}}}{a b^{3}}}}\right )}{32 \, {\left (a^{2} b^{3} c - \sqrt {a b} a^{2} b^{2} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {{\left (2 \, a^{2} b c d^{3} {\left | b \right |} - {\left (2 \, \sqrt {a b} b c^{2} d - 5 \, \sqrt {a b} a d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (4 \, a b^{2} c^{3} d - 5 \, a^{2} b c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b^{3} c - \sqrt {a^{2} b^{6} c^{2} - {\left (a b^{3} c^{2} - a^{2} b^{2} d^{2}\right )} a b^{3}}}{a b^{3}}}}\right )}{32 \, {\left (a^{2} b^{3} d + \sqrt {a b} a b^{3} c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {2 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} c d - 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d + 6 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d - 2 \, \sqrt {d x + c} b^{2} c^{4} d + 9 \, {\left (d x + c\right )}^{\frac {5}{2}} a b d^{3} - 16 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c d^{3} + 7 \, \sqrt {d x + c} a b c^{2} d^{3} - 5 \, \sqrt {d x + c} a^{2} d^{5}}{16 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )}^{2} a b^{2}} \] Input:

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

Output:

-1/32*(2*sqrt(a*b)*a^2*c*d^3*abs(b) + (2*a*b*c^2*d - 5*a^2*d^3)*abs(a)*abs 
(b)*abs(d) - (4*sqrt(a*b)*a*b*c^3*d - 5*sqrt(a*b)*a^2*c*d^3)*abs(b))*arcta 
n(sqrt(d*x + c)/sqrt(-(a*b^3*c + sqrt(a^2*b^6*c^2 - (a*b^3*c^2 - a^2*b^2*d 
^2)*a*b^3))/(a*b^3)))/((a^2*b^3*c - sqrt(a*b)*a^2*b^2*d)*sqrt(-b^2*c - sqr 
t(a*b)*b*d)*abs(a)*abs(d)) + 1/32*(2*a^2*b*c*d^3*abs(b) - (2*sqrt(a*b)*b*c 
^2*d - 5*sqrt(a*b)*a*d^3)*abs(a)*abs(b)*abs(d) - (4*a*b^2*c^3*d - 5*a^2*b* 
c*d^3)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^3*c - sqrt(a^2*b^6*c^2 - (a 
*b^3*c^2 - a^2*b^2*d^2)*a*b^3))/(a*b^3)))/((a^2*b^3*d + sqrt(a*b)*a*b^3*c) 
*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(a)*abs(d)) + 1/16*(2*(d*x + c)^(7/2)*b^2 
*c*d - 6*(d*x + c)^(5/2)*b^2*c^2*d + 6*(d*x + c)^(3/2)*b^2*c^3*d - 2*sqrt( 
d*x + c)*b^2*c^4*d + 9*(d*x + c)^(5/2)*a*b*d^3 - 16*(d*x + c)^(3/2)*a*b*c* 
d^3 + 7*sqrt(d*x + c)*a*b*c^2*d^3 - 5*sqrt(d*x + c)*a^2*d^5)/(((d*x + c)^2 
*b - 2*(d*x + c)*b*c + b*c^2 - a*d^2)^2*a*b^2)
 

Mupad [B] (verification not implemented)

Time = 13.97 (sec) , antiderivative size = 3696, normalized size of antiderivative = 13.89 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((c*d*(c + d*x)^(7/2))/(8*a) - ((8*a*c*d^3 - 3*b*c^3*d)*(c + d*x)^(3/2))/( 
8*a*b) - ((c + d*x)^(1/2)*(5*a^2*d^5 + 2*b^2*c^4*d - 7*a*b*c^2*d^3))/(16*a 
*b^2) + (3*d*(3*a*d^2 - 2*b*c^2)*(c + d*x)^(5/2))/(16*a*b))/(b^2*(c + d*x) 
^4 + a^2*d^4 + b^2*c^4 + (6*b^2*c^2 - 2*a*b*d^2)*(c + d*x)^2 - (4*b^2*c^3 
- 4*a*b*c*d^2)*(c + d*x) - 4*b^2*c*(c + d*x)^3 - 2*a*b*c^2*d^2) - atan(((( 
(10240*a^4*b^4*d^5 - 4096*a^3*b^5*c^2*d^3)/(2048*a^3*b^3) - 64*a*b^4*c*d^2 
*(c + d*x)^(1/2)*((16*a^3*b^7*c^5 - 25*a*d^5*(a^9*b^9)^(1/2) + 45*a^5*b^5* 
c*d^4 - 52*a^4*b^6*c^3*d^2 + 16*b*c^2*d^3*(a^9*b^9)^(1/2))/(4096*(a^6*b^10 
*c^2 - a^7*b^9*d^2)))^(1/2))*((16*a^3*b^7*c^5 - 25*a*d^5*(a^9*b^9)^(1/2) + 
 45*a^5*b^5*c*d^4 - 52*a^4*b^6*c^3*d^2 + 16*b*c^2*d^3*(a^9*b^9)^(1/2))/(40 
96*(a^6*b^10*c^2 - a^7*b^9*d^2)))^(1/2) + ((c + d*x)^(1/2)*(25*a^2*d^6 + 1 
6*b^2*c^4*d^2 - 36*a*b*c^2*d^4))/(64*a^2*b))*((16*a^3*b^7*c^5 - 25*a*d^5*( 
a^9*b^9)^(1/2) + 45*a^5*b^5*c*d^4 - 52*a^4*b^6*c^3*d^2 + 16*b*c^2*d^3*(a^9 
*b^9)^(1/2))/(4096*(a^6*b^10*c^2 - a^7*b^9*d^2)))^(1/2)*1i - (((10240*a^4* 
b^4*d^5 - 4096*a^3*b^5*c^2*d^3)/(2048*a^3*b^3) + 64*a*b^4*c*d^2*(c + d*x)^ 
(1/2)*((16*a^3*b^7*c^5 - 25*a*d^5*(a^9*b^9)^(1/2) + 45*a^5*b^5*c*d^4 - 52* 
a^4*b^6*c^3*d^2 + 16*b*c^2*d^3*(a^9*b^9)^(1/2))/(4096*(a^6*b^10*c^2 - a^7* 
b^9*d^2)))^(1/2))*((16*a^3*b^7*c^5 - 25*a*d^5*(a^9*b^9)^(1/2) + 45*a^5*b^5 
*c*d^4 - 52*a^4*b^6*c^3*d^2 + 16*b*c^2*d^3*(a^9*b^9)^(1/2))/(4096*(a^6*b^1 
0*c^2 - a^7*b^9*d^2)))^(1/2) - ((c + d*x)^(1/2)*(25*a^2*d^6 + 16*b^2*c^...
 

Reduce [B] (verification not implemented)

Time = 3.27 (sec) , antiderivative size = 2012, normalized size of antiderivative = 7.56 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(14*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)))*a**3*b*c*d**2 - 8*sqrt(a)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c 
)))*a**2*b**2*c**3 - 28*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)))*a**2*b**2*c*d**2*x**2 
+ 16*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)))*a*b**3*c**3*x**2 + 14*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)))*a*b**3*c*d**2*x**4 - 8*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**4*c**3*x**4 
 + 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**4*d**3 - 4*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**3*b*c**2*d - 20*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**3*b*d**3*x**2 + 8*sqrt 
(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqr 
t(b)*sqrt(a)*d - b*c)))*a**2*b**2*c**2*d*x**2 + 10*sqrt(b)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b* 
c)))*a**2*b**2*d**3*x**4 - 4*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((s 
qrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**3*c**2*d*...