\(\int \frac {(c+d x)^{5/2}}{x^2 (a-b x^2)} \, dx\) [582]

Optimal result

Integrand size = 23, antiderivative size = 181 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=-\frac {c^2 \sqrt {c+d x}}{a x}-\frac {5 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a^{3/2} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a^{3/2} b^{3/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.37 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=\frac {-\frac {\sqrt {a} c^2 \sqrt {c+d x}}{x}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^2 \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^3 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-5 \sqrt {a} c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 27, 1610, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*d*(-1/2*(c^2*Sqrt[c + d*x])/(a*d*x) - (5*c^(3/2)*ArcTanh[Sqrt[c + d*x]/S 
qrt[c]])/(2*a) - ((Sqrt[b]*c - Sqrt[a]*d)^(5/2)*ArcTanh[(b^(1/4)*Sqrt[c + 
d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^(3/2)*b^(3/4)*d) + ((Sqrt[b]*c + 
Sqrt[a]*d)^(5/2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]* 
d]])/(2*a^(3/2)*b^(3/4)*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 1610

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

rule 2009

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.36

method	result	size
risch	\(-\frac {c^{2} \sqrt {d x +c}}{a x}-\frac {d \left (2 b \left (\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+5 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )\right )}{a}\)	\(247\)
derivativedivides	\(-2 d^{3} \left (\frac {c^{2} \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a \,d^{2}}+\frac {b \left (\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}\right )\)	\(258\)
default	\(2 d^{3} \left (-\frac {c^{2} \left (\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a \,d^{2}}+\frac {b \left (\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}\right )\)	\(261\)
pseudoelliptic	\(\frac {3 d \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, x \left (-\frac {\sqrt {a b \,d^{2}}\, a \,d^{2}}{3}-\sqrt {a b \,d^{2}}\, b \,c^{2}+a b c \,d^{2}+\frac {c^{3} b^{2}}{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+3 \left (d x \left (\left (\frac {a \,d^{2}}{3}+b \,c^{2}\right ) \sqrt {a b \,d^{2}}+b c \left (a \,d^{2}+\frac {b \,c^{2}}{3}\right )\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (5 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) d x +c^{2} \sqrt {d x +c}\right )}{3}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, a \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, x}\)	\(283\)

Input:

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

Output:

-c^2*(d*x+c)^(1/2)/a/x-1/a*d*(2*b*(1/2*(-3*a*b*c*d^2-c^3*b^2+(a*b*d^2)^(1/ 
2)*a*d^2+3*(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))-1/2*(3 
*a*b*c*d^2+c^3*b^2+(a*b*d^2)^(1/2)*a*d^2+3*(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)))+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. 1647 vs. \(2 (133) = 266\).

Time = 2.45 (sec) , antiderivative size = 3302, normalized size of antiderivative = 18.24 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=- \int \frac {c^{2} \sqrt {c + d x}}{- a x^{2} + b x^{4}}\, dx - \int \frac {d^{2} x^{2} \sqrt {c + d x}}{- a x^{2} + b x^{4}}\, dx - \int \frac {2 c d x \sqrt {c + d x}}{- a x^{2} + b x^{4}}\, dx \] Input:

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

Output:

-Integral(c**2*sqrt(c + d*x)/(-a*x**2 + b*x**4), x) - Integral(d**2*x**2*s 
qrt(c + d*x)/(-a*x**2 + b*x**4), x) - Integral(2*c*d*x*sqrt(c + d*x)/(-a*x 
**2 + b*x**4), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (133) = 266\).

Time = 0.18 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.29 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=\frac {5 \, c^{2} d \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {\sqrt {d x + c} c^{2}}{a x} - \frac {{\left ({\left (3 \, b c^{2} d + a d^{3}\right )} a^{2} d^{2} {\left | b \right |} - 2 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (a b^{2} c^{4} d + 3 \, a^{2} b c^{2} d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a^{2} b d - \sqrt {a b} a b c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} - \frac {{\left ({\left (3 \, b c^{2} d + a d^{3}\right )} a^{2} d^{2} {\left | b \right |} + 2 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (a b^{2} c^{4} d + 3 \, a^{2} b c^{2} d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{{\left (a^{2} b d + \sqrt {a b} a b c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 7934, normalized size of antiderivative = 43.83 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((8*(4*a^7*b^2*c*d^17 - 92*a^3*b^6*c^9*d^9 - 488*a^4*b^5*c^7*d^11 + 
 176*a^5*b^4*c^5*d^13 + 400*a^6*b^3*c^3*d^15))/a^3 + (((8*(128*a^6*b^5*c^4 
*d^9 - 128*a^7*b^4*c^2*d^11))/a^3 - (16*(32*a^7*b^4*d^10 - 48*a^6*b^5*c^2* 
d^8)*(c + d*x)^(1/2)*((a^2*d^5*(a^7*b^3)^(1/2) + a^3*b^4*c^5 + 5*a^5*b^2*c 
*d^4 + 10*a^4*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^3)^(1/2) + 10*a*b*c^2*d^3*( 
a^7*b^3)^(1/2))/(4*a^6*b^3))^(1/2))/a^2)*((a^2*d^5*(a^7*b^3)^(1/2) + a^3*b 
^4*c^5 + 5*a^5*b^2*c*d^4 + 10*a^4*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^3)^(1/2 
) + 10*a*b*c^2*d^3*(a^7*b^3)^(1/2))/(4*a^6*b^3))^(1/2) + (16*(c + d*x)^(1/ 
2)*(76*a^6*b^3*c*d^14 - 20*a^3*b^6*c^7*d^8 - 304*a^4*b^5*c^5*d^10 + 20*a^5 
*b^4*c^3*d^12))/a^2)*((a^2*d^5*(a^7*b^3)^(1/2) + a^3*b^4*c^5 + 5*a^5*b^2*c 
*d^4 + 10*a^4*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^3)^(1/2) + 10*a*b*c^2*d^3*( 
a^7*b^3)^(1/2))/(4*a^6*b^3))^(1/2))*((a^2*d^5*(a^7*b^3)^(1/2) + a^3*b^4*c^ 
5 + 5*a^5*b^2*c*d^4 + 10*a^4*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^3)^(1/2) + 1 
0*a*b*c^2*d^3*(a^7*b^3)^(1/2))/(4*a^6*b^3))^(1/2) + (16*(c + d*x)^(1/2)*(2 
*a^6*b*d^20 + 2*b^7*c^12*d^8 + 13*a*b^6*c^10*d^10 + 405*a^2*b^5*c^8*d^12 + 
 335*a^3*b^4*c^6*d^14 + 55*a^4*b^3*c^4*d^16 - 12*a^5*b^2*c^2*d^18))/a^2)*( 
(a^2*d^5*(a^7*b^3)^(1/2) + a^3*b^4*c^5 + 5*a^5*b^2*c*d^4 + 10*a^4*b^3*c^3* 
d^2 + 5*b^2*c^4*d*(a^7*b^3)^(1/2) + 10*a*b*c^2*d^3*(a^7*b^3)^(1/2))/(4*a^6 
*b^3))^(1/2)*1i - (((8*(4*a^7*b^2*c*d^17 - 92*a^3*b^6*c^9*d^9 - 488*a^4*b^ 
5*c^7*d^11 + 176*a^5*b^4*c^5*d^13 + 400*a^6*b^3*c^3*d^15))/a^3 + (((8*(...
 

Reduce [B] (verification not implemented)

Time = 11.13 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.55 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a \,d^{2} x -2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b \,c^{2} x +4 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a c d x -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,d^{2} x -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b \,c^{2} x +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,d^{2} x +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b \,c^{2} x -2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a c d x +2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a c d x -2 \sqrt {d x +c}\, a b \,c^{2}+5 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b c d x -5 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b c d x}{2 a^{2} b x} \] Input:

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

Output:

( - 2*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*d**2*x - 2*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*c**2*x + 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*c*d*x - sqrt(a)*sqrt(sqrt(b)*s 
qrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d* 
x))*a*d**2*x - sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*s 
qrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c**2*x + sqrt(a)*sqrt(sqrt(b)*s 
qrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x)) 
*a*d**2*x + 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*c**2*x - 2*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*c*d*x + 2*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*c*d*x - 2*sqrt(c + d*x)*a*b*c**2 + 5* 
sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*b*c*d*x - 5*sqrt(c)*log(sqrt(c + d* 
x) + sqrt(c))*a*b*c*d*x)/(2*a**2*b*x)