\(\int \frac {1}{(c+d x) (a+b x^2)^{5/2}} \, dx\) [1282]

Optimal result

Integrand size = 19, antiderivative size = 154 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {a d+b c x}{3 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )^{3/2}}+\frac {3 a^2 d^3+b c \left (2 b c^2+5 a d^2\right ) x}{3 a^2 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {d^4 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {4 a^3 d^3+2 b^3 c^3 x^3+a^2 b d \left (c^2+6 c d x+3 d^2 x^2\right )+a b^2 c x \left (3 c^2+5 d^2 x^2\right )}{3 a^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}-\frac {2 d^4 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

Integrate[1/((c + d*x)*(a + b*x^2)^(5/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {496, 25, 686, 27, 488, 219}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 496

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

Int[1/((c + d*x)*(a + b*x^2)^(5/2)),x]
 

Output:

(a*d + b*c*x)/(3*a*(b*c^2 + a*d^2)*(a + b*x^2)^(3/2)) + ((3*a^2*d^3 + b*c* 
(2*b*c^2 + 5*a*d^2)*x)/(a*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) - (3*a*d^4*ArcT 
anh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(b*c^2 + a*d^2)^ 
(3/2))/(3*a*(b*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(600\) vs. \(2(140)=280\).

Time = 0.34 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.90

method	result	size
default	\(\frac {\frac {d^{2}}{3 \left (a \,d^{2}+b \,c^{2}\right ) \left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}+\frac {b c d \left (\frac {\frac {4 b \left (x +\frac {c}{d}\right )}{3}-\frac {4 b c}{3 d}}{\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}+\frac {16 b \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{3 {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right )}^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {d^{2} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}}{d}\)	\(601\)

Input:

int(1/(d*x+c)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (141) = 282\).

Time = 0.18 (sec) , antiderivative size = 858, normalized size of antiderivative = 5.57 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(a^2*b^2*d^4*x^4 + 2*a^3*b*d^4*x^2 + a^4*d^4)*sqrt(b*c^2 + a*d^2)* 
log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqr 
t(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) 
 + 2*(a^2*b^2*c^4*d + 5*a^3*b*c^2*d^3 + 4*a^4*d^5 + (2*b^4*c^5 + 7*a*b^3*c 
^3*d^2 + 5*a^2*b^2*c*d^4)*x^3 + 3*(a^2*b^2*c^2*d^3 + a^3*b*d^5)*x^2 + 3*(a 
*b^3*c^5 + 3*a^2*b^2*c^3*d^2 + 2*a^3*b*c*d^4)*x)*sqrt(b*x^2 + a))/(a^4*b^3 
*c^6 + 3*a^5*b^2*c^4*d^2 + 3*a^6*b*c^2*d^4 + a^7*d^6 + (a^2*b^5*c^6 + 3*a^ 
3*b^4*c^4*d^2 + 3*a^4*b^3*c^2*d^4 + a^5*b^2*d^6)*x^4 + 2*(a^3*b^4*c^6 + 3* 
a^4*b^3*c^4*d^2 + 3*a^5*b^2*c^2*d^4 + a^6*b*d^6)*x^2), -1/3*(3*(a^2*b^2*d^ 
4*x^4 + 2*a^3*b*d^4*x^2 + a^4*d^4)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 
 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a* 
b*d^2)*x^2)) - (a^2*b^2*c^4*d + 5*a^3*b*c^2*d^3 + 4*a^4*d^5 + (2*b^4*c^5 + 
 7*a*b^3*c^3*d^2 + 5*a^2*b^2*c*d^4)*x^3 + 3*(a^2*b^2*c^2*d^3 + a^3*b*d^5)* 
x^2 + 3*(a*b^3*c^5 + 3*a^2*b^2*c^3*d^2 + 2*a^3*b*c*d^4)*x)*sqrt(b*x^2 + a) 
)/(a^4*b^3*c^6 + 3*a^5*b^2*c^4*d^2 + 3*a^6*b*c^2*d^4 + a^7*d^6 + (a^2*b^5* 
c^6 + 3*a^3*b^4*c^4*d^2 + 3*a^4*b^3*c^2*d^4 + a^5*b^2*d^6)*x^4 + 2*(a^3*b^ 
4*c^6 + 3*a^4*b^3*c^4*d^2 + 3*a^5*b^2*c^2*d^4 + a^6*b*d^6)*x^2)]
 

Sympy [F]

\[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )}\, dx \] Input:

integrate(1/(d*x+c)/(b*x**2+a)**(5/2),x)
 

Output:

Integral(1/((a + b*x**2)**(5/2)*(c + d*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (141) = 282\).

Time = 0.06 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {b c x}{3 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} a b c^{2} + {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{2}\right )}} + \frac {b c x}{2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} + \frac {\sqrt {b x^{2} + a} a b^{2} c^{4}}{d^{2}} + \sqrt {b x^{2} + a} a^{3} d^{2}} + \frac {2 \, b c x}{3 \, {\left (\sqrt {b x^{2} + a} a^{2} b c^{2} + \sqrt {b x^{2} + a} a^{3} d^{2}\right )}} + \frac {1}{\frac {\sqrt {b x^{2} + a} b^{2} c^{4}}{d^{3}} + \frac {2 \, \sqrt {b x^{2} + a} a b c^{2}}{d} + \sqrt {b x^{2} + a} a^{2} d} + \frac {1}{3 \, {\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{2}}{d} + {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d\right )}} + \frac {\operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d} \] Input:

integrate(1/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/3*b*c*x/((b*x^2 + a)^(3/2)*a*b*c^2 + (b*x^2 + a)^(3/2)*a^2*d^2) + b*c*x/ 
(2*sqrt(b*x^2 + a)*a^2*b*c^2 + sqrt(b*x^2 + a)*a*b^2*c^4/d^2 + sqrt(b*x^2 
+ a)*a^3*d^2) + 2/3*b*c*x/(sqrt(b*x^2 + a)*a^2*b*c^2 + sqrt(b*x^2 + a)*a^3 
*d^2) + 1/(sqrt(b*x^2 + a)*b^2*c^4/d^3 + 2*sqrt(b*x^2 + a)*a*b*c^2/d + sqr 
t(b*x^2 + a)*a^2*d) + 1/3/((b*x^2 + a)^(3/2)*b*c^2/d + (b*x^2 + a)^(3/2)*a 
*d) + arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c) 
))/((a + b*c^2/d^2)^(5/2)*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (141) = 282\).

Time = 0.15 (sec) , antiderivative size = 998, normalized size of antiderivative = 6.48 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

-2*d^4*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - 
a*d^2))/((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*sqrt(-b*c^2 - a*d^2)) + 1/3*( 
(((2*b^10*c^15 + 17*a*b^9*c^13*d^2 + 60*a^2*b^8*c^11*d^4 + 115*a^3*b^7*c^9 
*d^6 + 130*a^4*b^6*c^7*d^8 + 87*a^5*b^5*c^5*d^10 + 32*a^6*b^4*c^3*d^12 + 5 
*a^7*b^3*c*d^14)*x/(a^2*b^9*c^16 + 8*a^3*b^8*c^14*d^2 + 28*a^4*b^7*c^12*d^ 
4 + 56*a^5*b^6*c^10*d^6 + 70*a^6*b^5*c^8*d^8 + 56*a^7*b^4*c^6*d^10 + 28*a^ 
8*b^3*c^4*d^12 + 8*a^9*b^2*c^2*d^14 + a^10*b*d^16) + 3*(a^2*b^8*c^12*d^3 + 
 6*a^3*b^7*c^10*d^5 + 15*a^4*b^6*c^8*d^7 + 20*a^5*b^5*c^6*d^9 + 15*a^6*b^4 
*c^4*d^11 + 6*a^7*b^3*c^2*d^13 + a^8*b^2*d^15)/(a^2*b^9*c^16 + 8*a^3*b^8*c 
^14*d^2 + 28*a^4*b^7*c^12*d^4 + 56*a^5*b^6*c^10*d^6 + 70*a^6*b^5*c^8*d^8 + 
 56*a^7*b^4*c^6*d^10 + 28*a^8*b^3*c^4*d^12 + 8*a^9*b^2*c^2*d^14 + a^10*b*d 
^16))*x + 3*(a*b^9*c^15 + 8*a^2*b^8*c^13*d^2 + 27*a^3*b^7*c^11*d^4 + 50*a^ 
4*b^6*c^9*d^6 + 55*a^5*b^5*c^7*d^8 + 36*a^6*b^4*c^5*d^10 + 13*a^7*b^3*c^3* 
d^12 + 2*a^8*b^2*c*d^14)/(a^2*b^9*c^16 + 8*a^3*b^8*c^14*d^2 + 28*a^4*b^7*c 
^12*d^4 + 56*a^5*b^6*c^10*d^6 + 70*a^6*b^5*c^8*d^8 + 56*a^7*b^4*c^6*d^10 + 
 28*a^8*b^3*c^4*d^12 + 8*a^9*b^2*c^2*d^14 + a^10*b*d^16))*x + (a^2*b^8*c^1 
4*d + 10*a^3*b^7*c^12*d^3 + 39*a^4*b^6*c^10*d^5 + 80*a^5*b^5*c^8*d^7 + 95* 
a^6*b^4*c^6*d^9 + 66*a^7*b^3*c^4*d^11 + 25*a^8*b^2*c^2*d^13 + 4*a^9*b*d^15 
)/(a^2*b^9*c^16 + 8*a^3*b^8*c^14*d^2 + 28*a^4*b^7*c^12*d^4 + 56*a^5*b^6*c^ 
10*d^6 + 70*a^6*b^5*c^8*d^8 + 56*a^7*b^4*c^6*d^10 + 28*a^8*b^3*c^4*d^12...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/((a + b*x^2)^(5/2)*(c + d*x)),x)
 

Output:

int(1/((a + b*x^2)^(5/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 3779, normalized size of antiderivative = 24.54 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(d*x+c)/(b*x^2+a)^(5/2),x)
 

Output:

( - 6*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)* 
sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt( 
b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**3*c*d**2 - 12*sqrt(b)* 
sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + 
b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 
 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b*c*d**2*x**2 - 6*sqrt(b)*sqrt(2*s 
qrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2)* 
atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c** 
2)*c - a*d**2 - 2*b*c**2))*a*b**2*c*d**2*x**4 - 6*sqrt(2*sqrt(b)*sqrt(a*d* 
*2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x 
)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**4*d**4 - 
 6*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt( 
a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d* 
*2 - 2*b*c**2))*a**3*b*c**2*d**2 - 12*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2) 
*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqr 
t(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**3*b*d**4*x**2 - 12*s 
qrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + 
b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 
 2*b*c**2))*a**2*b**2*c**2*d**2*x**2 - 6*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c* 
*2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt...