\(\int \frac {1}{x (c+d x)^{5/2} (a-b x^2)^3} \, dx\) [739]

Optimal result

Integrand size = 23, antiderivative size = 449 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\frac {2}{3 c \left (a-\frac {b c^2}{d^2}\right )^3 (c+d x)^{3/2}}-\frac {2 d^6 \left (7 b c^2-a d^2\right )}{c^2 \left (b c^2-a d^2\right )^4 \sqrt {c+d x}}-\frac {b^2 \sqrt {c+d x} \left (2 a b c^2 d^2 (14 c-43 d x)+a^2 d^4 (68 c-21 d x)-b^2 c^4 (8 c-19 d x)\right )}{16 a^2 \left (b c^2-a d^2\right )^4 \left (a-b x^2\right )}+\frac {b^2 \sqrt {c+d x} \left (b c^2 (c-3 d x)+a d^2 (3 c-d x)\right )}{4 a \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )^2}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{5/2}}+\frac {b^{5/4} \left (32 b c^2-114 \sqrt {a} \sqrt {b} c d+117 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^3 \left (\sqrt {b} c-\sqrt {a} d\right )^{9/2}}+\frac {b^{5/4} \left (32 b c^2+114 \sqrt {a} \sqrt {b} c d+117 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^3 \left (\sqrt {b} c+\sqrt {a} d\right )^{9/2}} \] Output:

2/3/c/(a-b*c^2/d^2)^3/(d*x+c)^(3/2)-2*d^6*(-a*d^2+7*b*c^2)/c^2/(-a*d^2+b*c 
^2)^4/(d*x+c)^(1/2)-1/16*b^2*(d*x+c)^(1/2)*(2*a*b*c^2*d^2*(-43*d*x+14*c)+a 
^2*d^4*(-21*d*x+68*c)-b^2*c^4*(-19*d*x+8*c))/a^2/(-a*d^2+b*c^2)^4/(-b*x^2+ 
a)+1/4*b^2*(d*x+c)^(1/2)*(b*c^2*(-3*d*x+c)+a*d^2*(-d*x+3*c))/a/(-a*d^2+b*c 
^2)^3/(-b*x^2+a)^2-2*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(5/2)+1/32*b^(5/ 
4)*(32*b*c^2-114*a^(1/2)*b^(1/2)*c*d+117*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1 
/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^3/(b^(1/2)*c-a^(1/2)*d)^(9/2)+1/32*b^(5 
/4)*(32*b*c^2+114*a^(1/2)*b^(1/2)*c*d+117*a*d^2)*arctanh(b^(1/4)*(d*x+c)^( 
1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^3/(b^(1/2)*c+a^(1/2)*d)^(9/2)
 

Mathematica [A] (verified)

Time = 3.79 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=-\frac {\frac {2 a \left (3 b^5 c^6 x^2 (8 c-19 d x) (c+d x)^2-32 a^5 d^8 (4 c+3 d x)-3 a b^4 c^4 (c+d x)^2 \left (12 c^3-31 c^2 d x+28 c d^2 x^2-86 d^3 x^3\right )+32 a^4 b d^6 \left (22 c^3+21 c^2 d x+8 c d^2 x^2+6 d^3 x^3\right )-a^3 b^2 d^4 \left (-240 c^5-405 c^4 d x+1318 c^3 d^2 x^2+1419 c^2 d^3 x^3+128 c d^4 x^4+96 d^5 x^5\right )+a^2 b^3 c^2 d^2 \left (60 c^5-162 c^4 d x-708 c^3 d^2 x^2-627 c^2 d^3 x^3+626 c d^4 x^4+735 d^5 x^5\right )\right )}{c^2 \left (b c^2-a d^2\right )^4 (c+d x)^{3/2} \left (a-b x^2\right )^2}+\frac {3 b \sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (32 b c^2+114 \sqrt {a} \sqrt {b} c d+117 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^5}-\frac {3 b^{3/2} \left (32 b c^2-114 \sqrt {a} \sqrt {b} c d+117 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^4 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}+\frac {192 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{5/2}}}{96 a^3} \] Input:

Integrate[1/(x*(c + d*x)^(5/2)*(a - b*x^2)^3),x]
 

Output:

-1/96*((2*a*(3*b^5*c^6*x^2*(8*c - 19*d*x)*(c + d*x)^2 - 32*a^5*d^8*(4*c + 
3*d*x) - 3*a*b^4*c^4*(c + d*x)^2*(12*c^3 - 31*c^2*d*x + 28*c*d^2*x^2 - 86* 
d^3*x^3) + 32*a^4*b*d^6*(22*c^3 + 21*c^2*d*x + 8*c*d^2*x^2 + 6*d^3*x^3) - 
a^3*b^2*d^4*(-240*c^5 - 405*c^4*d*x + 1318*c^3*d^2*x^2 + 1419*c^2*d^3*x^3 
+ 128*c*d^4*x^4 + 96*d^5*x^5) + a^2*b^3*c^2*d^2*(60*c^5 - 162*c^4*d*x - 70 
8*c^3*d^2*x^2 - 627*c^2*d^3*x^3 + 626*c*d^4*x^4 + 735*d^5*x^5)))/(c^2*(b*c 
^2 - a*d^2)^4*(c + d*x)^(3/2)*(a - b*x^2)^2) + (3*b*Sqrt[-(b*c) - Sqrt[a]* 
Sqrt[b]*d]*(32*b*c^2 + 114*Sqrt[a]*Sqrt[b]*c*d + 117*a*d^2)*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)^5 - (3*b^(3/2)*(32*b*c^2 - 114*Sqrt[a]*Sqrt[b]*c*d + 117 
*a*d^2)*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)^4*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d 
]) + (192*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(5/2))/a^3
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1253\) vs. \(2(449)=898\).

Time = 4.85 (sec) , antiderivative size = 1253, normalized size of antiderivative = 2.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {561, 25, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1674

\(\displaystyle -2 \int \left (-\frac {d^6}{c \left (b c^2-a d^2\right )^3 (c+d x)^2}+\frac {b^2 \left (\left (b c^2+a d^2\right ) (c+d x)-c \left (b c^2+3 a d^2\right )\right ) d^4}{a \left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^3}+\frac {b^2 \left (c \left (b^2 c^4-3 a b d^2 c^2-6 a^2 d^4\right )-\left (b^2 c^4-3 a b d^2 c^2-2 a^2 d^4\right ) (c+d x)\right ) d^2}{a^2 \left (b c^2-a d^2\right )^3 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^2}+\frac {a d^8-7 b c^2 d^6}{c^2 \left (b c^2-a d^2\right )^4 (c+d x)}+\frac {b^2 \left (\left (b^3 c^6-4 a b^2 d^2 c^4+6 a^2 b d^4 c^2+3 a^3 d^6\right ) (c+d x)-c \left (b^3 c^6-4 a b^2 d^2 c^4+6 a^2 b d^4 c^2+9 a^3 d^6\right )\right )}{a^3 \left (b c^2-a d^2\right )^4 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {1}{a^3 c^2 x d}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {\left (7 b c^2-a d^2\right ) d^6}{c^2 \left (b c^2-a d^2\right )^4 \sqrt {c+d x}}+\frac {d^6}{3 c \left (b c^2-a d^2\right )^3 (c+d x)^{3/2}}-\frac {b^2 \sqrt {c+d x} \left (4 c \left (b c^2+a d^2\right )-\left (3 b c^2+a d^2\right ) (c+d x)\right ) d^4}{8 a \left (b c^2-a d^2\right )^3 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^2}+\frac {b^2 \sqrt {c+d x} \left (c \left (11 b^2 c^4-34 a b d^2 c^2-25 a^2 d^4\right )-\left (11 b^2 c^4-30 a b d^2 c^2-5 a^2 d^4\right ) (c+d x)\right ) d^2}{32 a^2 \left (b c^2-a d^2\right )^4 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}+\frac {b^2 \sqrt {c+d x} \left (2 c \left (b^2 c^4-5 a b d^2 c^2-4 a^2 d^4\right )-\left (b^2 c^4-7 a b d^2 c^2-2 a^2 d^4\right ) (c+d x)\right ) d^2}{4 a^2 \left (b c^2-a d^2\right )^4 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {b^{5/4} \left (22 b^{3/2} c^3-55 \sqrt {a} b d c^2+40 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right ) d}{64 a^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2} \left (b c^2-a d^2\right )^2}+\frac {b^{5/4} \left (b^2 c^4+5 a b d^2 c^2-12 a^{3/2} \sqrt {b} d^3 c-2 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right ) d}{8 a^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \left (b c^2-a d^2\right )^3}+\frac {b^{5/4} \left (22 b^{3/2} c^3+55 \sqrt {a} b d c^2+40 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right ) d}{64 a^{5/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2} \left (b c^2-a d^2\right )^2}-\frac {b^{5/4} \left (b^2 c^4+5 a b d^2 c^2+12 a^{3/2} \sqrt {b} d^3 c-2 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right ) d}{8 a^{5/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \left (b c^2-a d^2\right )^3}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{5/2}}-\frac {b^{5/4} \left (b^3 c^6-4 a b^2 d^2 c^4+6 a^2 b d^4 c^2+6 a^{5/2} \sqrt {b} d^5 c+3 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^3 \sqrt {\sqrt {b} c-\sqrt {a} d} \left (b c^2-a d^2\right )^4}-\frac {b^{5/4} \left (b^3 c^6-4 a b^2 d^2 c^4+6 a^2 b d^4 c^2-6 a^{5/2} \sqrt {b} d^5 c+3 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 a^3 \sqrt {\sqrt {b} c+\sqrt {a} d} \left (b c^2-a d^2\right )^4}\right )\)

Input:

Int[1/(x*(c + d*x)^(5/2)*(a - b*x^2)^3),x]
 

Output:

-2*(d^6/(3*c*(b*c^2 - a*d^2)^3*(c + d*x)^(3/2)) + (d^6*(7*b*c^2 - a*d^2))/ 
(c^2*(b*c^2 - a*d^2)^4*Sqrt[c + d*x]) - (b^2*d^4*Sqrt[c + d*x]*(4*c*(b*c^2 
 + a*d^2) - (3*b*c^2 + a*d^2)*(c + d*x)))/(8*a*(b*c^2 - a*d^2)^3*(b*c^2 - 
a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)^2) + (b^2*d^2*Sqrt[c + d*x]*(c*(1 
1*b^2*c^4 - 34*a*b*c^2*d^2 - 25*a^2*d^4) - (11*b^2*c^4 - 30*a*b*c^2*d^2 - 
5*a^2*d^4)*(c + d*x)))/(32*a^2*(b*c^2 - a*d^2)^4*(b*c^2 - a*d^2 - 2*b*c*(c 
 + d*x) + b*(c + d*x)^2)) + (b^2*d^2*Sqrt[c + d*x]*(2*c*(b^2*c^4 - 5*a*b*c 
^2*d^2 - 4*a^2*d^4) - (b^2*c^4 - 7*a*b*c^2*d^2 - 2*a^2*d^4)*(c + d*x)))/(4 
*a^2*(b*c^2 - a*d^2)^4*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)) 
+ ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(a^3*c^(5/2)) - (b^(5/4)*d*(22*b^(3/2)*c^ 
3 - 55*Sqrt[a]*b*c^2*d + 40*a*Sqrt[b]*c*d^2 + 5*a^(3/2)*d^3)*ArcTanh[(b^(1 
/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(64*a^(5/2)*(Sqrt[b]*c - 
Sqrt[a]*d)^(5/2)*(b*c^2 - a*d^2)^2) + (b^(5/4)*d*(b^2*c^4 + 5*a*b*c^2*d^2 
- 12*a^(3/2)*Sqrt[b]*c*d^3 - 2*a^2*d^4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sq 
rt[Sqrt[b]*c - Sqrt[a]*d]])/(8*a^(5/2)*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)*(b*c^ 
2 - a*d^2)^3) - (b^(5/4)*(b^3*c^6 - 4*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + 6* 
a^(5/2)*Sqrt[b]*c*d^5 + 3*a^3*d^6)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sq 
rt[b]*c - Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c - Sqrt[a]*d]*(b*c^2 - a*d^2)^ 
4) + (b^(5/4)*d*(22*b^(3/2)*c^3 + 55*Sqrt[a]*b*c^2*d + 40*a*Sqrt[b]*c*d^2 
- 5*a^(3/2)*d^3)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[...
 

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 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 = 2.32 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(-2 d^{6} \left (-\frac {1}{3 c \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {a \,d^{2}-7 b \,c^{2}}{\left (a \,d^{2}-b \,c^{2}\right )^{4} c^{2} \sqrt {d x +c}}+\frac {b^{2} \left (\frac {\left (\frac {21}{32} a^{3} b \,d^{6}+\frac {43}{16} a^{2} b^{2} c^{2} d^{4}-\frac {19}{32} a \,b^{3} c^{4} d^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {a b c \,d^{2} \left (131 a^{2} d^{4}+286 b \,c^{2} d^{2} a -65 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {a \,d^{2} \left (25 a^{3} d^{6}-105 a^{2} b \,c^{2} d^{4}-345 a \,b^{2} c^{4} d^{2}+73 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {a c \,d^{2} \left (105 a^{3} d^{6}+25 a^{2} b \,c^{2} d^{4}-157 a \,b^{2} c^{4} d^{2}+27 b^{3} c^{6}\right ) \sqrt {d x +c}}{32}}{\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 {b \left (-\frac {\left (-354 d^{6} c \,a^{3} b +88 a^{2} c^{3} d^{4} b^{2}-14 a \,c^{5} d^{2} b^{3}+117 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+278 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}-147 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\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 (354 d^{6} c \,a^{3} b -88 a^{2} c^{3} d^{4} b^{2}+14 a \,c^{5} d^{2} b^{3}+117 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+278 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}-147 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\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}\right )}{a^{3} d^{6} \left (a \,d^{2}-b \,c^{2}\right )^{4}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}} a^{3} d^{6}}\right )\)	\(682\)
default	\(2 d^{6} \left (-\frac {-a \,d^{2}+7 b \,c^{2}}{\left (a \,d^{2}-b \,c^{2}\right )^{4} c^{2} \sqrt {d x +c}}+\frac {1}{3 c \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {b^{2} \left (\frac {\left (\frac {21}{32} a^{3} b \,d^{6}+\frac {43}{16} a^{2} b^{2} c^{2} d^{4}-\frac {19}{32} a \,b^{3} c^{4} d^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {a b c \,d^{2} \left (131 a^{2} d^{4}+286 b \,c^{2} d^{2} a -65 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {a \,d^{2} \left (25 a^{3} d^{6}-105 a^{2} b \,c^{2} d^{4}-345 a \,b^{2} c^{4} d^{2}+73 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {a c \,d^{2} \left (105 a^{3} d^{6}+25 a^{2} b \,c^{2} d^{4}-157 a \,b^{2} c^{4} d^{2}+27 b^{3} c^{6}\right ) \sqrt {d x +c}}{32}}{\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 {b \left (-\frac {\left (-354 d^{6} c \,a^{3} b +88 a^{2} c^{3} d^{4} b^{2}-14 a \,c^{5} d^{2} b^{3}+117 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+278 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}-147 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\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 (354 d^{6} c \,a^{3} b -88 a^{2} c^{3} d^{4} b^{2}+14 a \,c^{5} d^{2} b^{3}+117 \sqrt {a b \,d^{2}}\, a^{3} d^{6}+278 \sqrt {a b \,d^{2}}\, a^{2} b \,c^{2} d^{4}-147 \sqrt {a b \,d^{2}}\, a \,b^{2} c^{4} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{3} c^{6}\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}\right )}{a^{3} d^{6} \left (a \,d^{2}-b \,c^{2}\right )^{4}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}} a^{3} d^{6}}\right )\)	\(685\)
pseudoelliptic	\(\frac {-b^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\frac {117 a^{3} d^{6} c^{\frac {11}{2}}}{32}+\frac {139 c^{\frac {13}{2}} b \,a^{2} d^{5} x}{16}+\frac {117 a^{3} d^{7} x \,c^{\frac {9}{2}}}{32}+c^{\frac {15}{2}} b \left (c^{3} b^{2} \left (d x +c \right )-\frac {147 b c \,d^{2} \left (d x +c \right ) a}{32}+\frac {139 a^{2} d^{4}}{16}\right )\right ) \sqrt {a b \,d^{2}}+\frac {7 d^{2} \left (\frac {177 a^{2} d^{5} x \,c^{\frac {11}{2}}}{7}+\frac {177 a^{2} c^{\frac {13}{2}} d^{4}}{7}+c^{\frac {15}{2}} b \left (d x +c \right ) \left (b \,c^{2}-\frac {44 a \,d^{2}}{7}\right )\right ) b a}{16}\right ) \left (-b \,x^{2}+a \right )^{2} \sqrt {d x +c}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (\left (\left (\frac {117 a^{3} d^{6} c^{\frac {11}{2}}}{32}+\frac {139 c^{\frac {13}{2}} b \,a^{2} d^{5} x}{16}+\frac {117 a^{3} d^{7} x \,c^{\frac {9}{2}}}{32}+c^{\frac {15}{2}} b \left (c^{3} b^{2} \left (d x +c \right )-\frac {147 b c \,d^{2} \left (d x +c \right ) a}{32}+\frac {139 a^{2} d^{4}}{16}\right )\right ) \sqrt {a b \,d^{2}}-\frac {7 d^{2} \left (\frac {177 a^{2} d^{5} x \,c^{\frac {11}{2}}}{7}+\frac {177 a^{2} c^{\frac {13}{2}} d^{4}}{7}+c^{\frac {15}{2}} b \left (d x +c \right ) \left (b \,c^{2}-\frac {44 a \,d^{2}}{7}\right )\right ) b a}{16}\right ) b^{2} \left (-b \,x^{2}+a \right )^{2} \sqrt {d x +c}\, \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {3 \sqrt {a b \,d^{2}}\, \left (-\frac {8 \left (d x +c \right )^{\frac {3}{2}} \left (a \,d^{2}-b \,c^{2}\right )^{4} c^{2} \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{3}+\left (-\frac {176 d^{6} b \,a^{2} \left (\frac {313}{352} b^{2} x^{4}-\frac {659}{352} a b \,x^{2}+a^{2}\right ) c^{\frac {11}{2}}}{9}-\frac {45 d^{5} \left (\frac {86}{135} b^{2} x^{4}-\frac {209}{135} a b \,x^{2}+a^{2}\right ) x \,b^{2} a \,c^{\frac {13}{2}}}{4}-\frac {56 d^{7} x b \left (\frac {35}{32} b^{2} x^{4}-\frac {473}{224} a b \,x^{2}+a^{2}\right ) a^{2} c^{\frac {9}{2}}}{3}+\frac {8 d^{9} x \,a^{3} \left (-b \,x^{2}+a \right )^{2} c^{\frac {5}{2}}}{3}+\frac {32 a^{3} \left (-b \,x^{2}+a \right )^{2} d^{8} c^{\frac {7}{2}}}{9}+c^{\frac {15}{2}} \left (-\frac {20 a^{3} d^{4}}{3}-\frac {5 d^{2} \left (-\frac {59}{5} d^{2} x^{2}-\frac {27}{10} c d x +c^{2}\right ) b \,a^{2}}{3}+b^{2} \left (-\frac {7}{12} c^{3} d x -12 d^{4} x^{4}-\frac {61}{12} c \,d^{3} x^{3}-\frac {11}{6} d^{2} c^{2} x^{2}+c^{4}\right ) a -\frac {2 \left (d x +c \right )^{2} x^{2} b^{3} c \left (-\frac {19 d x}{8}+c \right )}{3}\right ) b^{2}\right ) a \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{4}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, c^{\frac {9}{2}} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (d x +c \right )^{\frac {3}{2}} \left (a \,d^{2}-b \,c^{2}\right )^{4} \left (-b \,x^{2}+a \right )^{2} a^{3}}\)	\(814\)

Input:

int(1/x/(d*x+c)^(5/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-2*d^6*(-1/3/c/(a*d^2-b*c^2)^3/(d*x+c)^(3/2)-(a*d^2-7*b*c^2)/(a*d^2-b*c^2) 
^4/c^2/(d*x+c)^(1/2)+b^2/a^3/d^6/(a*d^2-b*c^2)^4*(((21/32*a^3*b*d^6+43/16* 
a^2*b^2*c^2*d^4-19/32*a*b^3*c^4*d^2)*(d*x+c)^(7/2)-1/32*a*b*c*d^2*(131*a^2 
*d^4+286*a*b*c^2*d^2-65*b^2*c^4)*(d*x+c)^(5/2)-1/32*a*d^2*(25*a^3*d^6-105* 
a^2*b*c^2*d^4-345*a*b^2*c^4*d^2+73*b^3*c^6)*(d*x+c)^(3/2)+1/32*a*c*d^2*(10 
5*a^3*d^6+25*a^2*b*c^2*d^4-157*a*b^2*c^4*d^2+27*b^3*c^6)*(d*x+c)^(1/2))/(- 
b*(d*x+c)^2+2*b*c*(d*x+c)+a*d^2-b*c^2)^2+1/32*b*(-1/2*(-354*d^6*c*a^3*b+88 
*a^2*c^3*d^4*b^2-14*a*c^5*d^2*b^3+117*(a*b*d^2)^(1/2)*a^3*d^6+278*(a*b*d^2 
)^(1/2)*a^2*b*c^2*d^4-147*(a*b*d^2)^(1/2)*a*b^2*c^4*d^2+32*(a*b*d^2)^(1/2) 
*b^3*c^6)/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*(354*d^6*c*a^3*b-88*a^2*c^3 
*d^4*b^2+14*a*c^5*d^2*b^3+117*(a*b*d^2)^(1/2)*a^3*d^6+278*(a*b*d^2)^(1/2)* 
a^2*b*c^2*d^4-147*(a*b*d^2)^(1/2)*a*b^2*c^4*d^2+32*(a*b*d^2)^(1/2)*b^3*c^6 
)/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/c^(5/2)/a^3/d^6*arctanh((d*x+c)^(1 
/2)/c^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x/(d*x+c)^(5/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x/(d*x+c)**(5/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )}^{3} {\left (d x + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate(1/x/(d*x+c)^(5/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2704 vs. \(2 (384) = 768\).

Time = 0.62 (sec) , antiderivative size = 2704, normalized size of antiderivative = 6.02 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x/(d*x+c)^(5/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/32*((a^3*b^4*c^8*d - 4*a^4*b^3*c^6*d^3 + 6*a^5*b^2*c^4*d^5 - 4*a^6*b*c^ 
2*d^7 + a^7*d^9)^2*(32*b^3*c^6 - 147*a*b^2*c^4*d^2 + 278*a^2*b*c^2*d^4 + 1 
17*a^3*d^6)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b) - (32*sqrt(a*b)*a^2*b^7*c^ 
15 - 261*sqrt(a*b)*a^3*b^6*c^13*d^2 + 914*sqrt(a*b)*a^4*b^5*c^11*d^4 - 121 
5*sqrt(a*b)*a^5*b^4*c^9*d^6 - 180*sqrt(a*b)*a^6*b^3*c^7*d^8 + 1933*sqrt(a* 
b)*a^7*b^2*c^5*d^10 - 1694*sqrt(a*b)*a^8*b*c^3*d^12 + 471*sqrt(a*b)*a^9*c* 
d^14)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a^3*b^4*c^8*d - 4*a^4*b^3*c^6*d^3 + 
 6*a^5*b^2*c^4*d^5 - 4*a^6*b*c^2*d^7 + a^7*d^9)*abs(b) + 2*(7*a^6*b^11*c^2 
2*d^2 - 100*a^7*b^10*c^20*d^4 + 725*a^8*b^9*c^18*d^6 - 3040*a^9*b^8*c^16*d 
^8 + 7910*a^10*b^7*c^14*d^10 - 13384*a^11*b^6*c^12*d^12 + 15050*a^12*b^5*c 
^10*d^14 - 11200*a^13*b^4*c^8*d^16 + 5315*a^14*b^3*c^6*d^18 - 1460*a^15*b^ 
2*c^4*d^20 + 177*a^16*b*c^2*d^22)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b))*arc 
tan(sqrt(d*x + c)/sqrt(-(a^3*b^5*c^9 - 4*a^4*b^4*c^7*d^2 + 6*a^5*b^3*c^5*d 
^4 - 4*a^6*b^2*c^3*d^6 + a^7*b*c*d^8 + sqrt((a^3*b^5*c^9 - 4*a^4*b^4*c^7*d 
^2 + 6*a^5*b^3*c^5*d^4 - 4*a^6*b^2*c^3*d^6 + a^7*b*c*d^8)^2 - (a^3*b^5*c^1 
0 - 5*a^4*b^4*c^8*d^2 + 10*a^5*b^3*c^6*d^4 - 10*a^6*b^2*c^4*d^6 + 5*a^7*b* 
c^2*d^8 - a^8*d^10)*(a^3*b^5*c^8 - 4*a^4*b^4*c^6*d^2 + 6*a^5*b^3*c^4*d^4 - 
 4*a^6*b^2*c^2*d^6 + a^7*b*d^8)))/(a^3*b^5*c^8 - 4*a^4*b^4*c^6*d^2 + 6*a^5 
*b^3*c^4*d^4 - 4*a^6*b^2*c^2*d^6 + a^7*b*d^8)))/((sqrt(a*b)*a^5*b^9*c^18 - 
 9*sqrt(a*b)*a^6*b^8*c^16*d^2 + 36*sqrt(a*b)*a^7*b^7*c^14*d^4 - 84*sqrt...
 

Mupad [B] (verification not implemented)

Time = 20.62 (sec) , antiderivative size = 49272, normalized size of antiderivative = 109.74 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(-((((13689*a^6*d^13*(a^13*b^5)^(1/2) - 1024*a^6*b^9*c^13 - 96525*a^12 
*b^3*c*d^12 + 8316*a^7*b^8*c^11*d^2 - 27189*a^8*b^7*c^9*d^4 + 276*a^9*b^6* 
c^7*d^6 + 169146*a^10*b^5*c^5*d^8 - 366600*a^11*b^4*c^3*d^10 + 1920*b^6*c^ 
12*d*(a^13*b^5)^(1/2) - 18960*a*b^5*c^10*d^3*(a^13*b^5)^(1/2) + 273204*a^5 
*b*c^2*d^11*(a^13*b^5)^(1/2) + 93249*a^2*b^4*c^8*d^5*(a^13*b^5)^(1/2) - 20 
6316*a^3*b^3*c^6*d^7*(a^13*b^5)^(1/2) + 156814*a^4*b^2*c^4*d^9*(a^13*b^5)^ 
(1/2))/(4096*(a^21*d^18 - a^12*b^9*c^18 - 9*a^20*b*c^2*d^16 + 9*a^13*b^8*c 
^16*d^2 - 36*a^14*b^7*c^14*d^4 + 84*a^15*b^6*c^12*d^6 - 126*a^16*b^5*c^10* 
d^8 + 126*a^17*b^4*c^8*d^10 - 84*a^18*b^3*c^6*d^12 + 36*a^19*b^2*c^4*d^14) 
))^(1/2)*(((c + d*x)^(1/2)*(39582418599936*a^16*b^41*c^87*d^8 - 1431478240 
018432*a^17*b^40*c^85*d^10 + 25129256648441856*a^18*b^39*c^83*d^12 - 28437 
6587896356864*a^19*b^38*c^81*d^14 + 2324649262253604864*a^20*b^37*c^79*d^1 
6 - 14581422353696161792*a^21*b^36*c^77*d^18 + 72776463364462215168*a^22*b 
^35*c^75*d^20 - 295758213847361519616*a^23*b^34*c^73*d^22 + 99292939948095 
0718464*a^24*b^33*c^71*d^24 - 2775492056899014098944*a^25*b^32*c^69*d^26 + 
 6468232234562953936896*a^26*b^31*c^67*d^28 - 12476405694151079755776*a^27 
*b^30*c^65*d^30 + 19459965459596826378240*a^28*b^29*c^63*d^32 - 2298395803 
9971086991360*a^29*b^28*c^61*d^34 + 15815328608695363829760*a^30*b^27*c^59 
*d^36 + 8653243869544725872640*a^31*b^26*c^57*d^38 - 507701380265609999155 
20*a^32*b^25*c^55*d^40 + 100886684290577942446080*a^33*b^24*c^53*d^42 -...
 

Reduce [B] (verification not implemented)

Time = 67.41 (sec) , antiderivative size = 11911, normalized size of antiderivative = 26.53 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 702*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**5*b*c**4*d**7 - 702*s 
qrt(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**5*b*c**3*d**8*x - 3792*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**4*b**2*c**6*d**5 - 3792*sqrt(a)*sqrt 
(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sq 
rt(sqrt(b)*sqrt(a)*d - b*c)))*a**4*b**2*c**5*d**6*x + 1404*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**4*b**2*c**4*d**7*x**2 + 1404*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**4*b**2*c**3*d**8*x**3 + 1410*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**3*c**8*d**3 + 1410*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**3*c**7*d**4*x + 7584*sqrt(a)*sqrt(c + d*x)*s 
qrt(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**3*c**6*d**5*x**2 + 7584*sqrt(a)*sqrt(c + d*x)*s 
qrt(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**3*c**5*d**6*x**3 - 702*sqrt(a)*sqrt(c + d*x)...