\(\int \frac {1}{(c+d x)^{5/2} (a-b x^2)^2} \, dx\) [683]

Optimal result

Integrand size = 20, antiderivative size = 311 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=-\frac {d \left (3 b c^2+7 a d^2\right )}{6 a \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {b c d \left (b c^2+19 a d^2\right )}{2 a \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {a d-b c x}{2 a \left (b c^2-a d^2\right ) (c+d x)^{3/2} \left (a-b x^2\right )}-\frac {b^{3/4} \left (2 \sqrt {b} c-7 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 a^{3/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{7/2}}+\frac {b^{3/4} \left (2 \sqrt {b} c+7 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 a^{3/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{7/2}} \] Output:

-1/6*d*(7*a*d^2+3*b*c^2)/a/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)-1/2*b*c*d*(19*a* 
d^2+b*c^2)/a/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)-1/2*(-b*c*x+a*d)/a/(-a*d^2+b*c 
^2)/(d*x+c)^(3/2)/(-b*x^2+a)-1/4*b^(3/4)*(2*b^(1/2)*c-7*a^(1/2)*d)*arctanh 
(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(3/2)/(b^(1/2)*c-a^( 
1/2)*d)^(7/2)+1/4*b^(3/4)*(2*b^(1/2)*c+7*a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c 
)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(3/2)/(b^(1/2)*c+a^(1/2)*d)^(7/2)
 

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \left (4 a^3 d^5+3 b^3 c^3 x (c+d x)^2-a^2 b d^3 \left (55 c^2+54 c d x+7 d^2 x^2\right )+a b^2 c d \left (-9 c^3-9 c^2 d x+61 c d^2 x^2+57 d^3 x^3\right )\right )}{\left (b c^2-a d^2\right )^3 (c+d x)^{3/2} \left (-a+b x^2\right )}+\frac {3 b \left (2 \sqrt {b} c+7 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^3 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {3 b \left (2 \sqrt {b} c-7 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^3 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{12 a^{3/2}} \] Input:

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

Output:

((-2*Sqrt[a]*(4*a^3*d^5 + 3*b^3*c^3*x*(c + d*x)^2 - a^2*b*d^3*(55*c^2 + 54 
*c*d*x + 7*d^2*x^2) + a*b^2*c*d*(-9*c^3 - 9*c^2*d*x + 61*c*d^2*x^2 + 57*d^ 
3*x^3)))/((b*c^2 - a*d^2)^3*(c + d*x)^(3/2)*(-a + b*x^2)) + (3*b*(2*Sqrt[b 
]*c + 7*Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x]) 
/(Sqrt[b]*c + Sqrt[a]*d)])/((Sqrt[b]*c + Sqrt[a]*d)^3*Sqrt[-(b*c) - Sqrt[a 
]*Sqrt[b]*d]) - (3*b*(2*Sqrt[b]*c - 7*Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + Sqr 
t[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/((Sqrt[b]*c - Sqr 
t[a]*d)^3*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/(12*a^(3/2))
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.29, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {496, 27, 655, 25, 27, 655, 25, 654, 25, 27, 1480, 221}

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.

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

\(\Big \downarrow \) 496

\(\displaystyle \frac {\int \frac {2 b c^2+5 b d x c-7 a d^2}{2 (c+d x)^{5/2} \left (a-b x^2\right )}dx}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 b c^2+5 b d x c-7 a d^2}{(c+d x)^{5/2} \left (a-b x^2\right )}dx}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 655

\(\displaystyle \frac {-\frac {\int -\frac {b \left (2 c \left (b c^2-6 a d^2\right )+d \left (3 b c^2+7 a d^2\right ) x\right )}{(c+d x)^{3/2} \left (a-b x^2\right )}dx}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b \left (2 c \left (b c^2-6 a d^2\right )+d \left (3 b c^2+7 a d^2\right ) x\right )}{(c+d x)^{3/2} \left (a-b x^2\right )}dx}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \int \frac {2 c \left (b c^2-6 a d^2\right )+d \left (3 b c^2+7 a d^2\right ) x}{(c+d x)^{3/2} \left (a-b x^2\right )}dx}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 655

\(\displaystyle \frac {\frac {b \left (-\frac {\int -\frac {2 b^2 c^4-15 a b d^2 c^2+b d \left (b c^2+19 a d^2\right ) x c-7 a^2 d^4}{\sqrt {c+d x} \left (a-b x^2\right )}dx}{b c^2-a d^2}-\frac {2 c d \left (19 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {b \left (\frac {\int \frac {2 b^2 c^4-15 a b d^2 c^2+b d \left (b c^2+19 a d^2\right ) x c-7 a^2 d^4}{\sqrt {c+d x} \left (a-b x^2\right )}dx}{b c^2-a d^2}-\frac {2 c d \left (19 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {\frac {b \left (\frac {2 \int -\frac {d \left (b^2 c^4-34 a b d^2 c^2+b \left (b c^2+19 a d^2\right ) (c+d x) c-7 a^2 d^4\right )}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{b c^2-a d^2}-\frac {2 c d \left (19 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {b \left (-\frac {2 \int \frac {d \left (b^2 c^4-34 a b d^2 c^2+b \left (b c^2+19 a d^2\right ) (c+d x) c-7 a^2 d^4\right )}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{b c^2-a d^2}-\frac {2 c d \left (19 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (-\frac {2 d \int \frac {b^2 c^4-34 a b d^2 c^2+b \left (b c^2+19 a d^2\right ) (c+d x) c-7 a^2 d^4}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{b c^2-a d^2}-\frac {2 c d \left (19 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {b \left (-\frac {2 d \left (\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^3 \left (7 \sqrt {a} d+2 \sqrt {b} c\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\sqrt {b} \left (2 \sqrt {b} c-7 \sqrt {a} d\right ) \left (\sqrt {a} d+\sqrt {b} c\right )^3 \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )}{b c^2-a d^2}-\frac {2 c d \left (19 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{b c^2-a d^2}-\frac {2 d \left (7 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{4 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{2 a \left (a-b x^2\right ) (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(a*d - b*c*x)/(a*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*(a - b*x^2)) + ((-2* 
d*(3*b*c^2 + 7*a*d^2))/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + (b*((-2*c*d*( 
b*c^2 + 19*a*d^2))/((b*c^2 - a*d^2)*Sqrt[c + d*x]) - (2*d*(((2*Sqrt[b]*c - 
 7*Sqrt[a]*d)*(Sqrt[b]*c + Sqrt[a]*d)^3*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sq 
rt[Sqrt[b]*c - Sqrt[a]*d]])/(2*Sqrt[a]*b^(1/4)*d*Sqrt[Sqrt[b]*c - Sqrt[a]* 
d]) - ((Sqrt[b]*c - Sqrt[a]*d)^3*(2*Sqrt[b]*c + 7*Sqrt[a]*d)*ArcTanh[(b^(1 
/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^(1/4)*d*Sqrt 
[Sqrt[b]*c + Sqrt[a]*d])))/(b*c^2 - a*d^2)))/(b*c^2 - a*d^2))/(4*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 221

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

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 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 655

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

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(2 d^{3} \left (-\frac {1}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b c}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}+\frac {b \left (\frac {-\frac {b c \left (3 a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 a \,d^{2}}+\frac {\left (a^{2} d^{4}+6 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{4 a \,d^{2}}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (\frac {\left (7 a^{2} d^{4}+15 b \,c^{2} d^{2} a -2 b^{2} c^{4}+19 \sqrt {a b \,d^{2}}\, a c \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a^{2} d^{4}-15 b \,c^{2} d^{2} a +2 b^{2} c^{4}+19 \sqrt {a b \,d^{2}}\, a c \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,d^{2}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}\right )\)	\(402\)
default	\(2 d^{3} \left (-\frac {1}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b c}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}+\frac {b \left (\frac {-\frac {b c \left (3 a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 a \,d^{2}}+\frac {\left (a^{2} d^{4}+6 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{4 a \,d^{2}}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (\frac {\left (7 a^{2} d^{4}+15 b \,c^{2} d^{2} a -2 b^{2} c^{4}+19 \sqrt {a b \,d^{2}}\, a c \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a^{2} d^{4}-15 b \,c^{2} d^{2} a +2 b^{2} c^{4}+19 \sqrt {a b \,d^{2}}\, a c \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,d^{2}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}\right )\)	\(402\)
pseudoelliptic	\(\frac {\frac {7 d \left (d x +c \right )^{\frac {3}{2}} b^{2} \left (\frac {\left (19 a \,d^{2} c +b \,c^{3}\right ) \sqrt {a b \,d^{2}}}{7}+a^{2} d^{4}+\frac {15 b \,c^{2} d^{2} a}{7}-\frac {2 b^{2} c^{4}}{7}\right ) \left (-b \,x^{2}+a \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}+\frac {7 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d \left (d x +c \right )^{\frac {3}{2}} b^{2} \left (-b \,x^{2}+a \right ) \left (\frac {\left (-19 a \,d^{2} c -b \,c^{3}\right ) \sqrt {a b \,d^{2}}}{7}+a^{2} d^{4}+\frac {15 b \,c^{2} d^{2} a}{7}-\frac {2 b^{2} c^{4}}{7}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {8 \left (\frac {3 c^{3} x \left (d x +c \right )^{2} b^{3}}{4}-\frac {9 d \left (-\frac {19}{3} d^{3} x^{3}-\frac {61}{9} c \,d^{2} x^{2}+c^{2} d x +c^{3}\right ) a c \,b^{2}}{4}-\frac {55 d^{3} \left (\frac {7}{55} d^{2} x^{2}+\frac {54}{55} c d x +c^{2}\right ) a^{2} b}{4}+a^{3} d^{5}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{21}\right )}{4}}{\left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right )^{3} a \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d x +c \right )^{\frac {3}{2}}}\)	\(417\)

Input:

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

Output:

2*d^3*(-1/3/(a*d^2-b*c^2)^2/(d*x+c)^(3/2)+4/(a*d^2-b*c^2)^3*b*c/(d*x+c)^(1 
/2)+1/(a*d^2-b*c^2)^3*b*((-1/4*b*c*(3*a*d^2+b*c^2)/a/d^2*(d*x+c)^(3/2)+1/4 
*(a^2*d^4+6*a*b*c^2*d^2+b^2*c^4)/a/d^2*(d*x+c)^(1/2))/(-b*(d*x+c)^2+2*b*c* 
(d*x+c)+a*d^2-b*c^2)+1/4/a/d^2*b*(1/2*(7*a^2*d^4+15*b*c^2*d^2*a-2*b^2*c^4+ 
19*(a*b*d^2)^(1/2)*a*c*d^2+(a*b*d^2)^(1/2)*b*c^3)/(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*(-7*a^2*d^4-15*b*c^2*d^2*a+2*b^2*c^4+19*(a*b*d^2)^(1/2)*a*c*d^ 
2+(a*b*d^2)^(1/2)*b*c^3)/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*a 
rctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8308 vs. \(2 (248) = 496\).

Time = 2.98 (sec) , antiderivative size = 8308, normalized size of antiderivative = 26.71 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1998 vs. \(2 (248) = 496\).

Time = 0.34 (sec) , antiderivative size = 1998, normalized size of antiderivative = 6.42 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*((a*b^3*c^6*d - 3*a^2*b^2*c^4*d^3 + 3*a^3*b*c^2*d^5 - a^4*d^7)^2*(b^2* 
c^3*d + 19*a*b*c*d^3)*abs(b) + (sqrt(a*b)*b^5*c^10*d - 37*sqrt(a*b)*a*b^4* 
c^8*d^3 + 98*sqrt(a*b)*a^2*b^3*c^6*d^5 - 82*sqrt(a*b)*a^3*b^2*c^4*d^7 + 13 
*sqrt(a*b)*a^4*b*c^2*d^9 + 7*sqrt(a*b)*a^5*d^11)*abs(-a*b^3*c^6*d + 3*a^2* 
b^2*c^4*d^3 - 3*a^3*b*c^2*d^5 + a^4*d^7)*abs(b) - (2*a*b^9*c^17*d - 27*a^2 
*b^8*c^15*d^3 + 113*a^3*b^7*c^13*d^5 - 223*a^4*b^6*c^11*d^7 + 225*a^5*b^5* 
c^9*d^9 - 97*a^6*b^4*c^7*d^11 - 13*a^7*b^3*c^5*d^13 + 27*a^8*b^2*c^3*d^15 
- 7*a^9*b*c*d^17)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^4*c^7 - 3*a^2*b^ 
3*c^5*d^2 + 3*a^3*b^2*c^3*d^4 - a^4*b*c*d^6 + sqrt((a*b^4*c^7 - 3*a^2*b^3* 
c^5*d^2 + 3*a^3*b^2*c^3*d^4 - a^4*b*c*d^6)^2 - (a*b^4*c^8 - 4*a^2*b^3*c^6* 
d^2 + 6*a^3*b^2*c^4*d^4 - 4*a^4*b*c^2*d^6 + a^5*d^8)*(a*b^4*c^6 - 3*a^2*b^ 
3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 - a^4*b*d^6)))/(a*b^4*c^6 - 3*a^2*b^3*c^4*d^ 
2 + 3*a^3*b^2*c^2*d^4 - a^4*b*d^6)))/((a^2*b^6*c^12*d - 6*a^3*b^5*c^10*d^3 
 + 15*a^4*b^4*c^8*d^5 - 20*a^5*b^3*c^6*d^7 + 15*a^6*b^2*c^4*d^9 - 6*a^7*b* 
c^2*d^11 + a^8*d^13 + sqrt(a*b)*a*b^6*c^13 - 6*sqrt(a*b)*a^2*b^5*c^11*d^2 
+ 15*sqrt(a*b)*a^3*b^4*c^9*d^4 - 20*sqrt(a*b)*a^4*b^3*c^7*d^6 + 15*sqrt(a* 
b)*a^5*b^2*c^5*d^8 - 6*sqrt(a*b)*a^6*b*c^3*d^10 + sqrt(a*b)*a^7*c*d^12)*sq 
rt(-b^2*c + sqrt(a*b)*b*d)*abs(-a*b^3*c^6*d + 3*a^2*b^2*c^4*d^3 - 3*a^3*b* 
c^2*d^5 + a^4*d^7)) + 1/4*((a*b^3*c^6*d - 3*a^2*b^2*c^4*d^3 + 3*a^3*b*c^2* 
d^5 - a^4*d^7)^2*(b^2*c^3*d + 19*a*b*c*d^3)*abs(b) - (sqrt(a*b)*b^5*c^1...
 

Mupad [B] (verification not implemented)

Time = 12.94 (sec) , antiderivative size = 12290, normalized size of antiderivative = 39.52 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((((-(4*a^3*b^6*c^9 - 49*a^3*d^9*(a^9*b^3)^(1/2) + 315*a^7*b^2*c*d^8 
 - 63*a^4*b^5*c^7*d^2 + 189*a^5*b^4*c^5*d^4 + 1155*a^6*b^3*c^3*d^6 + 105*b 
^3*c^6*d^3*(a^9*b^3)^(1/2) - 819*a*b^2*c^4*d^5*(a^9*b^3)^(1/2) - 837*a^2*b 
*c^2*d^7*(a^9*b^3)^(1/2))/(64*(a^13*d^14 - a^6*b^7*c^14 - 7*a^12*b*c^2*d^1 
2 + 7*a^7*b^6*c^12*d^2 - 21*a^8*b^5*c^10*d^4 + 35*a^9*b^4*c^8*d^6 - 35*a^1 
0*b^3*c^6*d^8 + 21*a^11*b^2*c^4*d^10)))^(1/2)*((c + d*x)^(1/2)*(-(4*a^3*b^ 
6*c^9 - 49*a^3*d^9*(a^9*b^3)^(1/2) + 315*a^7*b^2*c*d^8 - 63*a^4*b^5*c^7*d^ 
2 + 189*a^5*b^4*c^5*d^4 + 1155*a^6*b^3*c^3*d^6 + 105*b^3*c^6*d^3*(a^9*b^3) 
^(1/2) - 819*a*b^2*c^4*d^5*(a^9*b^3)^(1/2) - 837*a^2*b*c^2*d^7*(a^9*b^3)^( 
1/2))/(64*(a^13*d^14 - a^6*b^7*c^14 - 7*a^12*b*c^2*d^12 + 7*a^7*b^6*c^12*d 
^2 - 21*a^8*b^5*c^10*d^4 + 35*a^9*b^4*c^8*d^6 - 35*a^10*b^3*c^6*d^8 + 21*a 
^11*b^2*c^4*d^10)))^(1/2)*(2048*a^21*b^4*c*d^32 - 2048*a^6*b^19*c^31*d^2 + 
 30720*a^7*b^18*c^29*d^4 - 215040*a^8*b^17*c^27*d^6 + 931840*a^9*b^16*c^25 
*d^8 - 2795520*a^10*b^15*c^23*d^10 + 6150144*a^11*b^14*c^21*d^12 - 1025024 
0*a^12*b^13*c^19*d^14 + 13178880*a^13*b^12*c^17*d^16 - 13178880*a^14*b^11* 
c^15*d^18 + 10250240*a^15*b^10*c^13*d^20 - 6150144*a^16*b^9*c^11*d^22 + 27 
95520*a^17*b^8*c^9*d^24 - 931840*a^18*b^7*c^7*d^26 + 215040*a^19*b^6*c^5*d 
^28 - 30720*a^20*b^5*c^3*d^30) - 1792*a^19*b^4*d^31 + 256*a^5*b^18*c^28*d^ 
3 - 11776*a^6*b^17*c^26*d^5 + 119552*a^7*b^16*c^24*d^7 - 609280*a^8*b^15*c 
^22*d^9 + 1923328*a^9*b^14*c^20*d^11 - 4116992*a^10*b^13*c^18*d^13 + 62...
 

Reduce [B] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 4569, normalized size of antiderivative = 14.69 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(156*sqrt(a)*sqrt(c + d*x)*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**4 + 156*sqrt 
(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sq 
rt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b*c*d**5*x + 96*sqrt(a)*sqrt(c 
+ d*x)*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**4*d**2 + 96*sqrt(a)*sqrt(c + d*x)* 
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**3*d**3*x - 156*sqrt(a)*sqrt(c + d*x)*sqrt 
(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)))*a**2*b**2*c**2*d**4*x**2 - 156*sqrt(a)*sqrt(c + d*x)*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*d**5*x**3 - 12*sqrt(a)*sqrt(c + d*x)*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**6 - 12*sqrt(a)*sqrt(c + d*x)*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**5*d*x - 96*sqrt(a)*sqrt(c + d*x)*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**4*d**2 
*x**2 - 96*sqrt(a)*sqrt(c + d*x)*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*d**3*x**3 
 + 12*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c ...