\(\int \frac {(c+d x)^{5/2}}{x^2 (a-b x^2)^3} \, dx\) [712]

Optimal result

Integrand size = 23, antiderivative size = 326 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=-\frac {c^2 \sqrt {c+d x}}{a^3 x}+\frac {\sqrt {c+d x} \left (b c^2 x+a d (2 c+d x)\right )}{4 a^2 \left (a-b x^2\right )^2}+\frac {\sqrt {c+d x} \left (14 b c^2 x+5 a d (3 c+d x)\right )}{16 a^3 \left (a-b x^2\right )}-\frac {5 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {5 \sqrt {\sqrt {b} c-\sqrt {a} d} \left (12 b c^2-10 \sqrt {a} \sqrt {b} c d+a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{7/2} b^{3/4}}+\frac {5 \sqrt {\sqrt {b} c+\sqrt {a} d} \left (12 b c^2+10 \sqrt {a} \sqrt {b} c d+a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{7/2} b^{3/4}} \] Output:

-c^2*(d*x+c)^(1/2)/a^3/x+1/4*(d*x+c)^(1/2)*(b*c^2*x+a*d*(d*x+2*c))/a^2/(-b 
*x^2+a)^2+1/16*(d*x+c)^(1/2)*(14*b*c^2*x+5*a*d*(d*x+3*c))/a^3/(-b*x^2+a)-5 
*c^(3/2)*d*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3-5/32*(b^(1/2)*c-a^(1/2)*d)^( 
1/2)*(12*b*c^2-10*a^(1/2)*b^(1/2)*c*d+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^(3/4)+5/32*(b^(1/2)*c+a^(1/2)*d)^( 
1/2)*(12*b*c^2+10*a^(1/2)*b^(1/2)*c*d+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^(3/4)
 

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c+d x} \left (30 b^2 c^2 x^4+a^2 \left (16 c^2-23 c d x-9 d^2 x^2\right )+5 a b x^2 \left (-10 c^2+3 c d x+d^2 x^2\right )\right )}{x \left (a-b x^2\right )^2}-\frac {5 \sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (12 b c^2+10 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b}+\frac {5 \sqrt {-b c+\sqrt {a} \sqrt {b} d} \left (12 b c^2-10 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{b}-160 \sqrt {a} c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{32 a^{7/2}} \] Input:

Integrate[(c + d*x)^(5/2)/(x^2*(a - b*x^2)^3),x]
 

Output:

((-2*Sqrt[a]*Sqrt[c + d*x]*(30*b^2*c^2*x^4 + a^2*(16*c^2 - 23*c*d*x - 9*d^ 
2*x^2) + 5*a*b*x^2*(-10*c^2 + 3*c*d*x + d^2*x^2)))/(x*(a - b*x^2)^2) - (5* 
Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*(12*b*c^2 + 10*Sqrt[a]*Sqrt[b]*c*d + a*d^ 
2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sq 
rt[a]*d)])/b + (5*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*(12*b*c^2 - 10*Sqrt[a]* 
Sqrt[b]*c*d + a*d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x 
])/(Sqrt[b]*c - Sqrt[a]*d)])/b - 160*Sqrt[a]*c^(3/2)*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. \(1027\) vs. \(2(326)=652\).

Time = 3.68 (sec) , antiderivative size = 1027, normalized size of antiderivative = 3.15, 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.

\(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1674

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (-\frac {\sqrt [4]{b} \left (2 b c^2-3 \sqrt {a} \sqrt {b} d c+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right ) c^2}{8 a^{7/2} d \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\sqrt [4]{b} \left (3-\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 ) c^2}{2 a^3 \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\sqrt [4]{b} \left (2 b c^2+3 \sqrt {a} \sqrt {b} d c+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right ) c^2}{8 a^{7/2} d \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}}+\frac {\sqrt [4]{b} \left (\frac {\sqrt {b} c}{\sqrt {a} d}+3\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right ) c^2}{2 a^3 \sqrt {\sqrt {b} c+\sqrt {a} d}}-\frac {\sqrt {c+d x} c^2}{2 a^3 d x}+\frac {b \sqrt {c+d x} \left (c \left (b c^2-5 a d^2\right )-\left (b c^2-3 a d^2\right ) (c+d x)\right ) c^2}{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 {5 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) c^{3/2}}{2 a^3}-\frac {\left (12 b^2 c^4-18 \sqrt {a} b^{3/2} d c^3+45 a b d^2 c^2-60 a^{3/2} \sqrt {b} d^3 c+5 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{64 a^{7/2} b^{3/4} d \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\left (12 b^2 c^4+18 \sqrt {a} b^{3/2} d c^3+45 a b d^2 c^2+60 a^{3/2} \sqrt {b} d^3 c+5 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{64 a^{7/2} b^{3/4} d \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}}+\frac {\sqrt {c+d x} \left (2 c \left (b c^2+a d^2\right ) \left (3 b c^2+5 a d^2\right )-\left (6 b^2 c^4+15 a b d^2 c^2-5 a^2 d^4\right ) (c+d x)\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 {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-\left (b^2 c^4-a^2 d^4\right ) (c+d x)\right )}{8 a^2 \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 )^2}\right )\)

Input:

Int[(c + d*x)^(5/2)/(x^2*(a - b*x^2)^3),x]
 

Output:

2*d*(-1/2*(c^2*Sqrt[c + d*x])/(a^3*d*x) - (d^2*Sqrt[c + d*x]*(c*(b*c^2 - a 
*d^2)^2 - (b^2*c^4 - a^2*d^4)*(c + d*x)))/(8*a^2*(b*c^2 - a*d^2)*(b*c^2 - 
a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)^2) + (b*c^2*Sqrt[c + d*x]*(c*(b*c 
^2 - 5*a*d^2) - (b*c^2 - 3*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]*(2*c*(b*c^2 
 + a*d^2)*(3*b*c^2 + 5*a*d^2) - (6*b^2*c^4 + 15*a*b*c^2*d^2 - 5*a^2*d^4)*( 
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)) - (5*c^(3/2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*a^3) + (b^(1/4 
)*c^2*(3 - (Sqrt[b]*c)/(Sqrt[a]*d))*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[S 
qrt[b]*c - Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - (b^(1/4)*c^2 
*(2*b*c^2 - 3*Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[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^2*c^4 - 18*Sqrt[a]*b^(3/2)*c^3*d + 45*a*b*c^2*d^2 - 60*a^(3/2) 
*Sqrt[b]*c*d^3 + 5*a^2*d^4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c 
 - Sqrt[a]*d]])/(64*a^(7/2)*b^(3/4)*d*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)) + (b^ 
(1/4)*c^2*(3 + (Sqrt[b]*c)/(Sqrt[a]*d))*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sq 
rt[Sqrt[b]*c + Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c + Sqrt[a]*d]) + (b^(1/4) 
*c^2*(2*b*c^2 + 3*Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[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^2*c^4 + 18*Sqrt[a]*b^(3/2)*c^3*d + 45*a*b*c^2*d^2 + 60*...
 

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 = 0.87 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {\frac {55 d x \left (a b c \,d^{2}+\frac {12 c^{3} b^{2}}{11}-\frac {\sqrt {a b \,d^{2}}\, a \,d^{2}}{11}-2 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \left (-b \,x^{2}+a \right )^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}+\frac {55 \left (d \left (\left (\frac {a \,d^{2}}{11}+2 b \,c^{2}\right ) \sqrt {a b \,d^{2}}+\left (a \,d^{2}+\frac {12 b \,c^{2}}{11}\right ) b c \right ) x \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {32 \left (5 d x \,c^{\frac {3}{2}} \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\left (\frac {5 b \left (a \,d^{2}+6 b \,c^{2}\right ) x^{4}}{16}+\frac {15 a b c d \,x^{3}}{16}+\left (-\frac {9}{16} a^{2} d^{2}-\frac {25}{8} a b \,c^{2}\right ) x^{2}-\frac {23 a^{2} c d x}{16}+a^{2} c^{2}\right ) \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{55}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{32}}{\left (-b \,x^{2}+a \right )^{2} a^{3} \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, x}\)	\(384\)
risch	\(-\frac {c^{2} \sqrt {d x +c}}{a^{3} x}-\frac {d \left (\frac {2 \left (\frac {5}{32} a b \,d^{2}+\frac {7}{16} b^{2} c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {21 b^{2} c^{3} \left (d x +c \right )^{\frac {5}{2}}}{8}+2 \left (-\frac {9}{32} a^{2} d^{4}-\frac {33}{32} b \,c^{2} d^{2} a +\frac {21}{16} b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}-\frac {7 c \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{8}}{\left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {5 b \left (-\frac {\left (11 a b c \,d^{2}+12 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-11 a b c \,d^{2}-12 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{16}+5 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )\right )}{a^{3}}\)	\(392\)
derivativedivides	\(-2 d^{7} \left (\frac {c^{2} \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{6}}-\frac {\frac {\left (-\frac {5}{32} a b \,d^{2}-\frac {7}{16} b^{2} c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}+\frac {21 b^{2} c^{3} \left (d x +c \right )^{\frac {5}{2}}}{16}+\left (\frac {9}{32} a^{2} d^{4}+\frac {33}{32} b \,c^{2} d^{2} a -\frac {21}{16} b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {7 c \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{16}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {5 b \left (-\frac {\left (-11 a b c \,d^{2}-12 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (11 a b c \,d^{2}+12 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{a^{3} d^{6}}\right )\)	\(409\)
default	\(2 d^{7} \left (-\frac {c^{2} \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{6}}+\frac {\frac {\left (-\frac {5}{32} a b \,d^{2}-\frac {7}{16} b^{2} c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}+\frac {21 b^{2} c^{3} \left (d x +c \right )^{\frac {5}{2}}}{16}+\left (\frac {9}{32} a^{2} d^{4}+\frac {33}{32} b \,c^{2} d^{2} a -\frac {21}{16} b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {7 c \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{16}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {5 b \left (-\frac {\left (-11 a b c \,d^{2}-12 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (11 a b c \,d^{2}+12 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}}{a^{3} d^{6}}\right )\)	\(409\)

Input:

int((d*x+c)^(5/2)/x^2/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

55/32*(d*x*(a*b*c*d^2+12/11*c^3*b^2-1/11*(a*b*d^2)^(1/2)*a*d^2-2*(a*b*d^2) 
^(1/2)*b*c^2)*(-b*x^2+a)^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))+(d*((1/11*a*d^2+2*b*c^2)*(a*b*d^ 
2)^(1/2)+(a*d^2+12/11*b*c^2)*b*c)*x*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/( 
(b*c+(a*b*d^2)^(1/2))*b)^(1/2))-32/55*(5*d*x*c^(3/2)*(-b*x^2+a)^2*arctanh( 
(d*x+c)^(1/2)/c^(1/2))+(5/16*b*(a*d^2+6*b*c^2)*x^4+15/16*a*b*c*d*x^3+(-9/1 
6*a^2*d^2-25/8*a*b*c^2)*x^2-23/16*a^2*c*d*x+a^2*c^2)*(d*x+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)/(a*b*d^2)^(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. 1915 vs. \(2 (260) = 520\).

Time = 1.95 (sec) , antiderivative size = 3839, normalized size of antiderivative = 11.78 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)/x^2/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(5/2)/x**2/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\int { -\frac {{\left (d x + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{3} x^{2}} \,d x } \] Input:

integrate((d*x+c)^(5/2)/x^2/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate((d*x + c)^(5/2)/((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. 614 vs. \(2 (260) = 520\).

Time = 0.31 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.88 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\frac {5 \, c^{2} d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {\sqrt {d x + c} c^{2}}{a^{3} x} - \frac {5 \, {\left ({\left (22 \, b c^{2} d + a d^{3}\right )} a^{2} d^{2} {\left | b \right |} - 10 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (12 \, a b^{2} c^{4} d + 11 \, a^{2} b c^{2} 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 {5 \, {\left ({\left (22 \, b c^{2} d + a d^{3}\right )} a^{2} d^{2} {\left | b \right |} + 10 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (12 \, a b^{2} c^{4} d + 11 \, a^{2} b c^{2} 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 {14 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} c^{2} d - 42 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{3} d + 42 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{4} d - 14 \, \sqrt {d x + c} b^{2} c^{5} d + 5 \, {\left (d x + c\right )}^{\frac {7}{2}} a b d^{3} - 33 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{3} + 28 \, \sqrt {d x + c} a b c^{3} d^{3} - 9 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} d^{5} - 14 \, \sqrt {d x + c} a^{2} c 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)^(5/2)/x^2/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

5*c^2*d*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - sqrt(d*x + c)*c^2/ 
(a^3*x) - 5/32*((22*b*c^2*d + a*d^3)*a^2*d^2*abs(b) - 10*(sqrt(a*b)*b*c^3* 
d - sqrt(a*b)*a*c*d^3)*abs(a)*abs(b)*abs(d) - (12*a*b^2*c^4*d + 11*a^2*b*c 
^2*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)) - 5/32*((22*b*c^2*d + a*d^3)*a^2*d 
^2*abs(b) + 10*(sqrt(a*b)*b*c^3*d - sqrt(a*b)*a*c*d^3)*abs(a)*abs(b)*abs(d 
) - (12*a*b^2*c^4*d + 11*a^2*b*c^2*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)) - 
1/16*(14*(d*x + c)^(7/2)*b^2*c^2*d - 42*(d*x + c)^(5/2)*b^2*c^3*d + 42*(d* 
x + c)^(3/2)*b^2*c^4*d - 14*sqrt(d*x + c)*b^2*c^5*d + 5*(d*x + c)^(7/2)*a* 
b*d^3 - 33*(d*x + c)^(3/2)*a*b*c^2*d^3 + 28*sqrt(d*x + c)*a*b*c^3*d^3 - 9* 
(d*x + c)^(3/2)*a^2*d^5 - 14*sqrt(d*x + c)*a^2*c*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 = 11.33 (sec) , antiderivative size = 8309, normalized size of antiderivative = 25.49 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^(5/2)/(x^2*(a - b*x^2)^3),x)
 

Output:

atan(((((5*(1600*a^11*b^2*c*d^17 - 34560000*a^7*b^6*c^9*d^9 - 21158400*a^8 
*b^5*c^7*d^11 + 51504000*a^9*b^4*c^5*d^13 + 4212800*a^10*b^3*c^3*d^15))/(6 
5536*a^13) + (((5*(14155776*a^14*b^5*c^4*d^9 - 14155776*a^15*b^4*c^2*d^11) 
)/(65536*a^13) - ((16777216*a^15*b^4*d^10 - 25165824*a^14*b^5*c^2*d^8)*(c 
+ d*x)^(1/2)*(-(25*(a^2*d^5*(a^15*b^3)^(1/2) - 144*a^7*b^4*c^5 - 21*a^9*b^ 
2*c*d^4 - 364*a^8*b^3*c^3*d^2 + 384*b^2*c^4*d*(a^15*b^3)^(1/2) + 144*a*b*c 
^2*d^3*(a^15*b^3)^(1/2)))/(4096*a^14*b^3))^(1/2))/(32768*a^10))*(-(25*(a^2 
*d^5*(a^15*b^3)^(1/2) - 144*a^7*b^4*c^5 - 21*a^9*b^2*c*d^4 - 364*a^8*b^3*c 
^3*d^2 + 384*b^2*c^4*d*(a^15*b^3)^(1/2) + 144*a*b*c^2*d^3*(a^15*b^3)^(1/2) 
))/(4096*a^14*b^3))^(1/2) + ((c + d*x)^(1/2)*(4249600*a^10*b^3*c*d^14 - 36 
864000*a^7*b^6*c^7*d^8 - 135782400*a^8*b^5*c^5*d^10 + 61798400*a^9*b^4*c^3 
*d^12))/(32768*a^10))*(-(25*(a^2*d^5*(a^15*b^3)^(1/2) - 144*a^7*b^4*c^5 - 
21*a^9*b^2*c*d^4 - 364*a^8*b^3*c^3*d^2 + 384*b^2*c^4*d*(a^15*b^3)^(1/2) + 
144*a*b*c^2*d^3*(a^15*b^3)^(1/2)))/(4096*a^14*b^3))^(1/2))*(-(25*(a^2*d^5* 
(a^15*b^3)^(1/2) - 144*a^7*b^4*c^5 - 21*a^9*b^2*c*d^4 - 364*a^8*b^3*c^3*d^ 
2 + 384*b^2*c^4*d*(a^15*b^3)^(1/2) + 144*a*b*c^2*d^3*(a^15*b^3)^(1/2)))/(4 
096*a^14*b^3))^(1/2) + ((c + d*x)^(1/2)*(625*a^6*b*d^20 + 12960000*b^7*c^1 
2*d^8 + 6480000*a*b^6*c^10*d^10 + 283470000*a^2*b^5*c^8*d^12 + 31445000*a^ 
3*b^4*c^6*d^14 + 4300625*a^4*b^3*c^4*d^16 - 96250*a^5*b^2*c^2*d^18))/(3276 
8*a^10))*(-(25*(a^2*d^5*(a^15*b^3)^(1/2) - 144*a^7*b^4*c^5 - 21*a^9*b^2...
 

Reduce [B] (verification not implemented)

Time = 25.56 (sec) , antiderivative size = 1606, normalized size of antiderivative = 4.93 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(5/2)/x^2/(-b*x^2+a)^3,x)
 

Output:

( - 10*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*d**2*x - 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)))*a**2*b*c**2*x + 20*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*d**2*x**3 + 2 
40*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**2*c**2*x**3 - 10*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**2*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**3*c**2*x**5 + 
 100*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*c*d*x - 200*sqrt(b)*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*c*d*x**3 + 100*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*b**2*c*d*x**5 - 5* 
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))*a**3*d**2*x - 60*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))*a**2* 
b*c**2*x + 10*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sq 
rt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a**2*b*d**2*x**3 + 120*sqrt(a)*...