\(\int \frac {(c+d x)^{3/2}}{x^2 (a-b x^2)^3} \, dx\) [703]

Optimal result

Integrand size = 23, antiderivative size = 306 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^3} \, dx=-\frac {15 c \sqrt {c+d x}}{8 a^3 x}+\frac {(c+d x)^{3/2}}{4 a x \left (a-b x^2\right )^2}+\frac {\sqrt {c+d x} (10 c+7 d x)}{16 a^2 x \left (a-b x^2\right )}-\frac {3 \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {3 \left (20 b c^2-26 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{7/2} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {3 \left (20 b c^2+26 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{7/2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

-15/8*c*(d*x+c)^(1/2)/a^3/x+1/4*(d*x+c)^(3/2)/a/x/(-b*x^2+a)^2+1/16*(d*x+c 
)^(1/2)*(7*d*x+10*c)/a^2/x/(-b*x^2+a)-3*c^(1/2)*d*arctanh((d*x+c)^(1/2)/c^ 
(1/2))/a^3-3/32*(20*b*c^2-26*a^(1/2)*b^(1/2)*c*d+7*a*d^2)*arctanh(b^(1/4)* 
(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(7/2)/b^(1/4)/(b^(1/2)*c-a^(1 
/2)*d)^(1/2)+3/32*(20*b*c^2+26*a^(1/2)*b^(1/2)*c*d+7*a*d^2)*arctanh(b^(1/4 
)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(7/2)/b^(1/4)/(b^(1/2)*c+a^ 
(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 2.34 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c+d x} \left (30 b^2 c x^4+a^2 (16 c-11 d x)+a b x^2 (-50 c+7 d x)\right )}{x \left (a-b x^2\right )^2}+\frac {3 \left (20 b c^2+26 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {3 \left (20 b c^2-26 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}-96 \sqrt {a} \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{32 a^{7/2}} \] Input:

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

Output:

((-2*Sqrt[a]*Sqrt[c + d*x]*(30*b^2*c*x^4 + a^2*(16*c - 11*d*x) + a*b*x^2*( 
-50*c + 7*d*x)))/(x*(a - b*x^2)^2) + (3*(20*b*c^2 + 26*Sqrt[a]*Sqrt[b]*c*d 
 + 7*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] - (3*(20*b*c^2 - 26*S 
qrt[a]*Sqrt[b]*c*d + 7*a*d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqr 
t[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d] - 9 
6*Sqrt[a]*Sqrt[c]*d*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(32*a^(7/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(940\) vs. \(2(306)=612\).

Time = 3.24 (sec) , antiderivative size = 940, normalized size of antiderivative = 3.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 27, 1674, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1674

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (-\frac {\sqrt {c+d x} \left (b c^2-b (c+d x) c-a d^2\right ) d^2}{8 a^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^2}-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 a^3}+\frac {\sqrt [4]{b} c \left (2-\frac {\sqrt {b} c}{\sqrt {a} d}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^3 \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\sqrt [4]{b} c \left (\frac {\sqrt {b} c}{\sqrt {a} d}+2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 a^3 \sqrt {\sqrt {b} c+\sqrt {a} d}}+\frac {b c \sqrt {c+d x} \left (c \left (b c^2-3 a d^2\right )-\left (b c^2-2 a d^2\right ) (c+d x)\right )}{4 a^3 \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}+\frac {\sqrt {c+d x} \left (6 b^2 c^4+3 a b d^2 c^2-2 b \left (3 b c^2+a d^2\right ) (c+d x) c+7 a^2 d^4\right )}{32 a^3 \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {\sqrt [4]{b} c \left (2 b c^2-3 \sqrt {a} \sqrt {b} d c+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{8 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} d}-\frac {\left (12 b^{3/2} c^3-18 \sqrt {a} b d c^2+19 a \sqrt {b} d^2 c-21 a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{64 a^{7/2} \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} d}+\frac {\sqrt [4]{b} c \left (2 b c^2+3 \sqrt {a} \sqrt {b} d c+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{8 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} d}+\frac {\left (12 b^{3/2} c^3+18 \sqrt {a} b d c^2+19 a \sqrt {b} d^2 c+21 a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{64 a^{7/2} \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} d}-\frac {c \sqrt {c+d x}}{2 a^3 x d}\right )\)

Input:

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

Output:

2*d*(-1/2*(c*Sqrt[c + d*x])/(a^3*d*x) - (d^2*Sqrt[c + d*x]*(b*c^2 - a*d^2 
- b*c*(c + d*x)))/(8*a^2*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2) 
^2) + (b*c*Sqrt[c + d*x]*(c*(b*c^2 - 3*a*d^2) - (b*c^2 - 2*a*d^2)*(c + d*x 
)))/(4*a^3*(b*c^2 - a*d^2)*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^ 
2)) + (Sqrt[c + d*x]*(6*b^2*c^4 + 3*a*b*c^2*d^2 + 7*a^2*d^4 - 2*b*c*(3*b*c 
^2 + a*d^2)*(c + d*x)))/(32*a^3*(b*c^2 - a*d^2)*(b*c^2 - a*d^2 - 2*b*c*(c 
+ d*x) + b*(c + d*x)^2)) - (3*Sqrt[c]*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*a 
^3) + (b^(1/4)*c*(2 - (Sqrt[b]*c)/(Sqrt[a]*d))*ArcTanh[(b^(1/4)*Sqrt[c + d 
*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - ( 
b^(1/4)*c*(2*b*c^2 - 3*Sqrt[a]*Sqrt[b]*c*d + 2*a*d^2)*ArcTanh[(b^(1/4)*Sqr 
t[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(8*a^(7/2)*d*(Sqrt[b]*c - Sqrt[a 
]*d)^(3/2)) - ((12*b^(3/2)*c^3 - 18*Sqrt[a]*b*c^2*d + 19*a*Sqrt[b]*c*d^2 - 
 21*a^(3/2)*d^3)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]* 
d]])/(64*a^(7/2)*b^(1/4)*d*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)) + (b^(1/4)*c*(2 
+ (Sqrt[b]*c)/(Sqrt[a]*d))*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c 
+ Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c + Sqrt[a]*d]) + (b^(1/4)*c*(2*b*c^2 + 
 3*Sqrt[a]*Sqrt[b]*c*d + 2*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqr 
t[b]*c + Sqrt[a]*d]])/(8*a^(7/2)*d*(Sqrt[b]*c + Sqrt[a]*d)^(3/2)) + ((12*b 
^(3/2)*c^3 + 18*Sqrt[a]*b*c^2*d + 19*a*Sqrt[b]*c*d^2 + 21*a^(3/2)*d^3)*Arc 
Tanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(64*a^(7/2)*...
 

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 1674

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && N 
eQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])
 

rule 2009

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

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {\frac {21 d x b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (a \,d^{2}+\frac {20 b \,c^{2}}{7}-\frac {26 \sqrt {a b \,d^{2}}\, c}{7}\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}+\frac {21 \left (d \left (a \,d^{2}+\frac {20 b \,c^{2}}{7}+\frac {26 \sqrt {a b \,d^{2}}\, c}{7}\right ) x b \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {32 \left (\left (\frac {15}{8} b^{2} c \,x^{4}+\frac {7}{16} a b d \,x^{3}-\frac {25}{8} a b c \,x^{2}-\frac {11}{16} a^{2} d x +a^{2} c \right ) \sqrt {d x +c}+3 d x \sqrt {c}\, \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{21}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{32}}{\left (-b \,x^{2}+a \right )^{2} a^{3} \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, x}\)	\(325\)
risch	\(-\frac {c \sqrt {d x +c}}{a^{3} x}-\frac {d \left (\frac {\frac {7 b^{2} c \left (d x +c \right )^{\frac {7}{2}}}{8}+\frac {7 \left (a \,d^{2}-6 b \,c^{2}\right ) b \left (d x +c \right )^{\frac {5}{2}}}{16}-\frac {b c \left (16 a \,d^{2}-21 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{8}+2 \left (-\frac {11}{32} a^{2} d^{4}+\frac {25}{32} b \,c^{2} d^{2} a -\frac {7}{16} b^{2} c^{4}\right ) \sqrt {d x +c}}{\left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {3 b \left (-\frac {\left (7 a \,d^{2}+20 b \,c^{2}+26 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-7 a \,d^{2}-20 b \,c^{2}+26 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{16}+3 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )\right )}{a^{3}}\)	\(329\)
derivativedivides	\(-2 d^{7} \left (\frac {c \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{6}}-\frac {\frac {-\frac {7 b^{2} c \left (d x +c \right )^{\frac {7}{2}}}{16}-\frac {7 \left (a \,d^{2}-6 b \,c^{2}\right ) b \left (d x +c \right )^{\frac {5}{2}}}{32}+\frac {b c \left (16 a \,d^{2}-21 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16}+\left (\frac {11}{32} a^{2} d^{4}-\frac {25}{32} b \,c^{2} d^{2} a +\frac {7}{16} b^{2} c^{4}\right ) \sqrt {d x +c}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {3 b \left (\frac {\left (7 a \,d^{2}+20 b \,c^{2}-26 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a \,d^{2}-20 b \,c^{2}-26 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{a^{3} d^{6}}\right )\)	\(344\)
default	\(2 d^{7} \left (-\frac {c \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{6}}+\frac {\frac {-\frac {7 b^{2} c \left (d x +c \right )^{\frac {7}{2}}}{16}-\frac {7 \left (a \,d^{2}-6 b \,c^{2}\right ) b \left (d x +c \right )^{\frac {5}{2}}}{32}+\frac {b c \left (16 a \,d^{2}-21 b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16}+\left (\frac {11}{32} a^{2} d^{4}-\frac {25}{32} b \,c^{2} d^{2} a +\frac {7}{16} b^{2} c^{4}\right ) \sqrt {d x +c}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {3 b \left (\frac {\left (7 a \,d^{2}+20 b \,c^{2}-26 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a \,d^{2}-20 b \,c^{2}-26 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{a^{3} d^{6}}\right )\)	\(344\)

Input:

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

Output:

21/32/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)*(d*x*b*((b*c+(a*b*d 
^2)^(1/2))*b)^(1/2)*(a*d^2+20/7*b*c^2-26/7*(a*b*d^2)^(1/2)*c)*(-b*x^2+a)^2 
*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+(d*(a*d^2+20/7*b 
*c^2+26/7*(a*b*d^2)^(1/2)*c)*x*b*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b* 
c+(a*b*d^2)^(1/2))*b)^(1/2))-32/21*((15/8*b^2*c*x^4+7/16*a*b*d*x^3-25/8*a* 
b*c*x^2-11/16*a^2*d*x+a^2*c)*(d*x+c)^(1/2)+3*d*x*c^(1/2)*(-b*x^2+a)^2*arct 
anh((d*x+c)^(1/2)/c^(1/2)))*(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)/x/(-b 
*x^2+a)^2/a^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2552 vs. \(2 (238) = 476\).

Time = 1.89 (sec) , antiderivative size = 5113, normalized size of antiderivative = 16.71 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (238) = 476\).

Time = 0.31 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\frac {3 \, c d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {3 \, {\left (26 \, a^{2} b c d^{3} {\left | b \right |} - {\left (6 \, \sqrt {a b} b c^{2} d - 7 \, \sqrt {a b} a d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (20 \, a b^{2} c^{3} d + 7 \, a^{2} b c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{3} b c + \sqrt {a^{6} b^{2} c^{2} - {\left (a^{3} b c^{2} - a^{4} d^{2}\right )} a^{3} b}}{a^{3} b}}}\right )}{32 \, {\left (a^{4} b d - \sqrt {a b} a^{3} b c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} - \frac {3 \, {\left (26 \, \sqrt {a b} a^{2} c d^{3} {\left | b \right |} + {\left (6 \, a b c^{2} d - 7 \, a^{2} d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (20 \, \sqrt {a b} a b c^{3} d + 7 \, \sqrt {a b} a^{2} c d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{3} b c - \sqrt {a^{6} b^{2} c^{2} - {\left (a^{3} b c^{2} - a^{4} d^{2}\right )} a^{3} b}}{a^{3} b}}}\right )}{32 \, {\left (a^{4} b c + \sqrt {a b} a^{4} d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} - \frac {\sqrt {d x + c} c}{a^{3} x} - \frac {14 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} c d - 42 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d + 42 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d - 14 \, \sqrt {d x + c} b^{2} c^{4} d + 7 \, {\left (d x + c\right )}^{\frac {5}{2}} a b d^{3} - 32 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c d^{3} + 25 \, \sqrt {d x + c} a b c^{2} d^{3} - 11 \, \sqrt {d x + c} a^{2} d^{5}}{16 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )}^{2} a^{3}} \] Input:

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

Output:

3*c*d*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - 3/32*(26*a^2*b*c*d^3 
*abs(b) - (6*sqrt(a*b)*b*c^2*d - 7*sqrt(a*b)*a*d^3)*abs(a)*abs(b)*abs(d) - 
 (20*a*b^2*c^3*d + 7*a^2*b*c*d^3)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a^3* 
b*c + sqrt(a^6*b^2*c^2 - (a^3*b*c^2 - a^4*d^2)*a^3*b))/(a^3*b)))/((a^4*b*d 
 - sqrt(a*b)*a^3*b*c)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a)*abs(d)) - 3/32*( 
26*sqrt(a*b)*a^2*c*d^3*abs(b) + (6*a*b*c^2*d - 7*a^2*d^3)*abs(a)*abs(b)*ab 
s(d) - (20*sqrt(a*b)*a*b*c^3*d + 7*sqrt(a*b)*a^2*c*d^3)*abs(b))*arctan(sqr 
t(d*x + c)/sqrt(-(a^3*b*c - sqrt(a^6*b^2*c^2 - (a^3*b*c^2 - a^4*d^2)*a^3*b 
))/(a^3*b)))/((a^4*b*c + sqrt(a*b)*a^4*d)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs 
(a)*abs(d)) - sqrt(d*x + c)*c/(a^3*x) - 1/16*(14*(d*x + c)^(7/2)*b^2*c*d - 
 42*(d*x + c)^(5/2)*b^2*c^2*d + 42*(d*x + c)^(3/2)*b^2*c^3*d - 14*sqrt(d*x 
 + c)*b^2*c^4*d + 7*(d*x + c)^(5/2)*a*b*d^3 - 32*(d*x + c)^(3/2)*a*b*c*d^3 
 + 25*sqrt(d*x + c)*a*b*c^2*d^3 - 11*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^3)
 

Mupad [B] (verification not implemented)

Time = 10.10 (sec) , antiderivative size = 7640, normalized size of antiderivative = 24.97 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(c^(1/2)*d*atan((b^5*c^(9/2)*d^13*(c + d*x)^(1/2)*21514977i)/(2048*((21514 
977*b^5*c^5*d^13)/2048 - (13433769*a*b^4*c^3*d^15)/4096 + (583443*a^2*b^3* 
c*d^17)/32768 - (2156625*b^6*c^7*d^11)/(256*a) + (151875*b^7*c^9*d^9)/(128 
*a^2))) - (b^4*c^(5/2)*d^15*(c + d*x)^(1/2)*13433769i)/(4096*((21514977*b^ 
5*c^5*d^13)/(2048*a) - (13433769*b^4*c^3*d^15)/4096 - (2156625*b^6*c^7*d^1 
1)/(256*a^2) + (151875*b^7*c^9*d^9)/(128*a^3) + (583443*a*b^3*c*d^17)/3276 
8)) - (b^6*c^(13/2)*d^11*(c + d*x)^(1/2)*2156625i)/(256*((21514977*a*b^5*c 
^5*d^13)/2048 - (2156625*b^6*c^7*d^11)/256 + (583443*a^3*b^3*c*d^17)/32768 
 - (13433769*a^2*b^4*c^3*d^15)/4096 + (151875*b^7*c^9*d^9)/(128*a))) + (b^ 
7*c^(17/2)*d^9*(c + d*x)^(1/2)*151875i)/(128*((151875*b^7*c^9*d^9)/128 - ( 
2156625*a*b^6*c^7*d^11)/256 + (583443*a^4*b^3*c*d^17)/32768 + (21514977*a^ 
2*b^5*c^5*d^13)/2048 - (13433769*a^3*b^4*c^3*d^15)/4096)) + (b^3*c^(1/2)*d 
^17*(c + d*x)^(1/2)*583443i)/(32768*((583443*b^3*c*d^17)/32768 - (13433769 
*b^4*c^3*d^15)/(4096*a) + (21514977*b^5*c^5*d^13)/(2048*a^2) - (2156625*b^ 
6*c^7*d^11)/(256*a^3) + (151875*b^7*c^9*d^9)/(128*a^4))))*3i)/a^3 - atan(( 
(((3*(15662592*a^8*b^5*c^4*d^11 - 35481600*a^7*b^6*c^6*d^9 + 18361728*a^9* 
b^4*c^2*d^13))/(65536*a^13) - (((3*(14942208*a^15*b^4*c*d^11 - 15204352*a^ 
14*b^5*c^3*d^9))/(65536*a^13) - ((16777216*a^15*b^4*d^10 - 25165824*a^14*b 
^5*c^2*d^8)*(c + d*x)^(1/2)*(-(9*(400*a^7*b^3*c^5 + 49*a^2*d^5*(a^15*b)^(1 
/2) - 640*b^2*c^4*d*(a^15*b)^(1/2) - 84*a^8*b^2*c^3*d^2 - 315*a^9*b*c*d...
 

Reduce [B] (verification not implemented)

Time = 26.03 (sec) , antiderivative size = 2380, normalized size of antiderivative = 7.78 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 114*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*x + 120*sqrt(a)*sqrt(sqr 
t(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**3*x + 228*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*a 
tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*c 
*d**2*x**3 - 240*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**3 - 114*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**5 + 120*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**5 + 42*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*x - 36*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*x - 84*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**3 + 72*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*b**2*c**2*d*x**3 + 42*sq 
rt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(s 
qrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*d**3*x**5 - 36*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 -...