\(\int \frac {(c+d x)^{5/2}}{x^2 (a-b x^2)^2} \, dx\) [658]

Optimal result

Integrand size = 23, antiderivative size = 262 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^2} \, dx=-\frac {c^2 \sqrt {c+d x}}{a^2 x}+\frac {\sqrt {c+d x} \left (b c^2 x+a d (2 c+d x)\right )}{2 a^2 \left (a-b x^2\right )}-\frac {5 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \left (6 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 a^{5/2} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \left (6 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 a^{5/2} b^{3/4}} \] Output:

-c^2*(d*x+c)^(1/2)/a^2/x+1/2*(d*x+c)^(1/2)*(b*c^2*x+a*d*(d*x+2*c))/a^2/(-b 
*x^2+a)-5*c^(3/2)*d*arctanh((d*x+c)^(1/2)/c^(1/2))/a^2-1/4*(b^(1/2)*c-a^(1 
/2)*d)^(3/2)*(6*b^(1/2)*c-a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2 
)*c-a^(1/2)*d)^(1/2))/a^(5/2)/b^(3/4)+1/4*(b^(1/2)*c+a^(1/2)*d)^(3/2)*(6*b 
^(1/2)*c+a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1 
/2))/a^(5/2)/b^(3/4)
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {c+d x} \left (-2 a c^2+2 a c d x+3 b c^2 x^2+a d^2 x^2\right )}{a x-b x^3}-\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (6 b c^2+7 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (6 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-20 \sqrt {a} c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^{5/2}} \] Input:

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

Output:

((2*Sqrt[a]*Sqrt[c + d*x]*(-2*a*c^2 + 2*a*c*d*x + 3*b*c^2*x^2 + a*d^2*x^2) 
)/(a*x - b*x^3) - (Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*(6*b*c^2 + 7*Sqrt[a]*S 
qrt[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 - ((Sqrt[b]*c - Sqrt[a]*d)^2*(6*Sqrt[b]*c - 
Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b 
]*c - Sqrt[a]*d)])/(Sqrt[b]*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) - 20*Sqrt[a] 
*c^(3/2)*d*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*a^(5/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(566\) vs. \(2(262)=524\).

Time = 1.86 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.16, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1674

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*d*(-1/2*(c^2*Sqrt[c + d*x])/(a^2*d*x) + (Sqrt[c + d*x]*(c*(b*c^2 - a*d^2 
)^2 - (b^2*c^4 - a^2*d^4)*(c + d*x)))/(4*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)) - (5*c^(3/2)*ArcTanh[Sqrt[c + d*x]/S 
qrt[c]])/(2*a^2) + (b^(1/4)*c^2*(3 - (Sqrt[b]*c)/(Sqrt[a]*d))*ArcTanh[(b^( 
1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^2*Sqrt[Sqrt[b]*c - 
Sqrt[a]*d]) - ((2*b^(3/2)*c^3 - Sqrt[a]*b*c^2*d + 8*a*Sqrt[b]*c*d^2 - a^(3 
/2)*d^3)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(8* 
a^(5/2)*b^(3/4)*d*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) + (b^(1/4)*c^2*(3 + (Sqrt[b 
]*c)/(Sqrt[a]*d))*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a] 
*d]])/(2*a^2*Sqrt[Sqrt[b]*c + Sqrt[a]*d]) + ((2*b^(3/2)*c^3 + Sqrt[a]*b*c^ 
2*d + 8*a*Sqrt[b]*c*d^2 + a^(3/2)*d^3)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqr 
t[Sqrt[b]*c + Sqrt[a]*d]])/(8*a^(5/2)*b^(3/4)*d*Sqrt[Sqrt[b]*c + Sqrt[a]*d 
]))
 

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 = 0.67 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.24

method	result	size
risch	\(-\frac {c^{2} \sqrt {d x +c}}{a^{2} x}-\frac {d \left (\frac {2 \left (\frac {a \,d^{2}}{4}+\frac {b \,c^{2}}{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+2 \left (\frac {1}{4} a \,d^{2} c -\frac {1}{4} b \,c^{3}\right ) \sqrt {d x +c}}{b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}}+\frac {b \left (-\frac {\left (8 a b c \,d^{2}+6 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+13 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-8 a b c \,d^{2}-6 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+13 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+5 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )\right )}{a^{2}}\)	\(326\)
pseudoelliptic	\(\frac {2 d x \left (\frac {\left (-a \,d^{2}-13 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{8}+\left (a \,d^{2}+\frac {3 b \,c^{2}}{4}\right ) b c \right ) \left (-b \,x^{2}+a \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+2 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d x \left (\frac {\left (a \,d^{2}+13 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{8}+\left (a \,d^{2}+\frac {3 b \,c^{2}}{4}\right ) b c \right ) \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {\left (5 d x \,c^{\frac {3}{2}} \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\left (\frac {\left (-a \,d^{2}-3 b \,c^{2}\right ) x^{2}}{2}-a d x c +a \,c^{2}\right ) \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{2}\right )}{a^{2} \left (-b \,x^{2}+a \right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, x}\)	\(340\)
derivativedivides	\(2 d^{5} \left (\frac {\frac {\left (\frac {a \,d^{2}}{4}+\frac {b \,c^{2}}{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {1}{4} a \,d^{2} c -\frac {1}{4} b \,c^{3}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-8 a b c \,d^{2}-6 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-13 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (8 a b c \,d^{2}+6 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-13 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{a^{2} d^{4}}-\frac {c^{2} \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{2} d^{4}}\right )\)	\(343\)
default	\(2 d^{5} \left (\frac {\frac {\left (\frac {a \,d^{2}}{4}+\frac {b \,c^{2}}{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {1}{4} a \,d^{2} c -\frac {1}{4} b \,c^{3}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-8 a b c \,d^{2}-6 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-13 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (8 a b c \,d^{2}+6 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-13 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{a^{2} d^{4}}-\frac {c^{2} \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{2} d^{4}}\right )\)	\(343\)

Input:

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

Output:

-c^2*(d*x+c)^(1/2)/a^2/x-1/a^2*d*(2*((1/4*a*d^2+1/4*b*c^2)*(d*x+c)^(3/2)+( 
1/4*a*d^2*c-1/4*b*c^3)*(d*x+c)^(1/2))/(b*(d*x+c)^2-2*b*c*(d*x+c)-a*d^2+b*c 
^2)+1/2*b*(-1/2*(8*a*b*c*d^2+6*c^3*b^2+(a*b*d^2)^(1/2)*a*d^2+13*(a*b*d^2)^ 
(1/2)*b*c^2)/b/(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*(-8*a*b*c*d^2-6*c^3*b^2+ 
(a*b*d^2)^(1/2)*a*d^2+13*(a*b*d^2)^(1/2)*b*c^2)/b/(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)))+5*c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1807 vs. \(2 (199) = 398\).

Time = 2.50 (sec) , antiderivative size = 3623, normalized size of antiderivative = 13.83 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (199) = 398\).

Time = 0.25 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.02 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^2} \, dx=\frac {5 \, c^{2} d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {{\left ({\left (13 \, a b c^{2} d + a^{2} d^{3}\right )} d^{2} {\left | b \right |} - 7 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | b \right |} {\left | d \right |} - 2 \, {\left (3 \, b^{2} c^{4} d + 4 \, a b c^{2} d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c + \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{4 \, {\left (a^{3} b d - \sqrt {a b} a^{2} b c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} - \frac {{\left ({\left (13 \, a b c^{2} d + a^{2} d^{3}\right )} d^{2} {\left | b \right |} + 7 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | b \right |} {\left | d \right |} - 2 \, {\left (3 \, b^{2} c^{4} d + 4 \, a b c^{2} d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{2} b c - \sqrt {a^{4} b^{2} c^{2} - {\left (a^{2} b c^{2} - a^{3} d^{2}\right )} a^{2} b}}{a^{2} b}}}\right )}{4 \, {\left (a^{3} b d + \sqrt {a b} a^{2} b c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} - \frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} b c^{2} d - 6 \, {\left (d x + c\right )}^{\frac {3}{2}} b c^{3} d + 3 \, \sqrt {d x + c} b c^{4} d + {\left (d x + c\right )}^{\frac {5}{2}} a d^{3} - 3 \, \sqrt {d x + c} a c^{2} d^{3}}{2 \, {\left ({\left (d x + c\right )}^{3} b - 3 \, {\left (d x + c\right )}^{2} b c + 3 \, {\left (d x + c\right )} b c^{2} - b c^{3} - {\left (d x + c\right )} a d^{2} + a c d^{2}\right )} a^{2}} \] Input:

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

Output:

5*c^2*d*arctan(sqrt(d*x + c)/sqrt(-c))/(a^2*sqrt(-c)) - 1/4*((13*a*b*c^2*d 
 + a^2*d^3)*d^2*abs(b) - 7*(sqrt(a*b)*b*c^3*d - sqrt(a*b)*a*c*d^3)*abs(b)* 
abs(d) - 2*(3*b^2*c^4*d + 4*a*b*c^2*d^3)*abs(b))*arctan(sqrt(d*x + c)/sqrt 
(-(a^2*b*c + sqrt(a^4*b^2*c^2 - (a^2*b*c^2 - a^3*d^2)*a^2*b))/(a^2*b)))/(( 
a^3*b*d - sqrt(a*b)*a^2*b*c)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(d)) - 1/4*(( 
13*a*b*c^2*d + a^2*d^3)*d^2*abs(b) + 7*(sqrt(a*b)*b*c^3*d - sqrt(a*b)*a*c* 
d^3)*abs(b)*abs(d) - 2*(3*b^2*c^4*d + 4*a*b*c^2*d^3)*abs(b))*arctan(sqrt(d 
*x + c)/sqrt(-(a^2*b*c - sqrt(a^4*b^2*c^2 - (a^2*b*c^2 - a^3*d^2)*a^2*b))/ 
(a^2*b)))/((a^3*b*d + sqrt(a*b)*a^2*b*c)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs( 
d)) - 1/2*(3*(d*x + c)^(5/2)*b*c^2*d - 6*(d*x + c)^(3/2)*b*c^3*d + 3*sqrt( 
d*x + c)*b*c^4*d + (d*x + c)^(5/2)*a*d^3 - 3*sqrt(d*x + c)*a*c^2*d^3)/(((d 
*x + c)^3*b - 3*(d*x + c)^2*b*c + 3*(d*x + c)*b*c^2 - b*c^3 - (d*x + c)*a* 
d^2 + a*c*d^2)*a^2)
 

Mupad [B] (verification not implemented)

Time = 9.00 (sec) , antiderivative size = 8051, normalized size of antiderivative = 30.73 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((8*a^9*b^2*c*d^17 - 26784*a^5*b^6*c^9*d^9 - 40056*a^6*b^5*c^7*d^11 
 + 57432*a^7*b^4*c^5*d^13 + 9400*a^8*b^3*c^3*d^15)/(16*a^8) + (((16896*a^1 
0*b^5*c^4*d^9 - 16896*a^11*b^4*c^2*d^11)/(16*a^8) - ((4096*a^11*b^4*d^10 - 
 6144*a^10*b^5*c^2*d^8)*(c + d*x)^(1/2)*(-(a^2*d^5*(a^11*b^3)^(1/2) - 36*a 
^5*b^4*c^5 - 15*a^7*b^2*c*d^4 - 145*a^6*b^3*c^3*d^2 + 120*b^2*c^4*d*(a^11* 
b^3)^(1/2) + 75*a*b*c^2*d^3*(a^11*b^3)^(1/2))/(64*a^10*b^3))^(1/2))/(8*a^6 
))*(-(a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4*c^5 - 15*a^7*b^2*c*d^4 - 145*a 
^6*b^3*c^3*d^2 + 120*b^2*c^4*d*(a^11*b^3)^(1/2) + 75*a*b*c^2*d^3*(a^11*b^3 
)^(1/2))/(64*a^10*b^3))^(1/2) + ((c + d*x)^(1/2)*(1888*a^8*b^3*c*d^14 - 57 
60*a^5*b^6*c^7*d^8 - 35232*a^6*b^5*c^5*d^10 + 13760*a^7*b^4*c^3*d^12))/(8* 
a^6))*(-(a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4*c^5 - 15*a^7*b^2*c*d^4 - 14 
5*a^6*b^3*c^3*d^2 + 120*b^2*c^4*d*(a^11*b^3)^(1/2) + 75*a*b*c^2*d^3*(a^11* 
b^3)^(1/2))/(64*a^10*b^3))^(1/2))*(-(a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4 
*c^5 - 15*a^7*b^2*c*d^4 - 145*a^6*b^3*c^3*d^2 + 120*b^2*c^4*d*(a^11*b^3)^( 
1/2) + 75*a*b*c^2*d^3*(a^11*b^3)^(1/2))/(64*a^10*b^3))^(1/2) + ((c + d*x)^ 
(1/2)*(a^6*b*d^20 + 1296*b^7*c^12*d^8 + 1944*a*b^6*c^10*d^10 + 61065*a^2*b 
^5*c^8*d^12 + 12380*a^3*b^4*c^6*d^14 + 1790*a^4*b^3*c^4*d^16 - 76*a^5*b^2* 
c^2*d^18))/(8*a^6))*(-(a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4*c^5 - 15*a^7* 
b^2*c*d^4 - 145*a^6*b^3*c^3*d^2 + 120*b^2*c^4*d*(a^11*b^3)^(1/2) + 75*a*b* 
c^2*d^3*(a^11*b^3)^(1/2))/(64*a^10*b^3))^(1/2)*1i - (((8*a^9*b^2*c*d^17...
 

Reduce [F]

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

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

Output:

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