\(\int \frac {1}{x (c+d x)^{3/2} (b c^2-b d^2 x^2)^{5/2}} \, dx\) [840]

Optimal result

Integrand size = 32, antiderivative size = 307 \[ \int \frac {1}{x (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\frac {1}{6 b c^2 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}+\frac {7}{16 b c^3 \sqrt {c+d x} \left (b c^2-b d^2 x^2\right )^{3/2}}+\frac {81 \sqrt {c+d x}}{64 b c^4 \left (b c^2-b d^2 x^2\right )^{3/2}}-\frac {277 (c+d x)^{3/2}}{384 b c^5 \left (b c^2-b d^2 x^2\right )^{3/2}}-\frac {21 \sqrt {c+d x}}{256 b^2 c^6 \sqrt {b c^2-b d^2 x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{b^{5/2} c^{13/2}}+\frac {533 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{256 \sqrt {2} b^{5/2} c^{13/2}} \] Output:

1/6/b/c^2/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(3/2)+7/16/b/c^3/(d*x+c)^(1/2)/ 
(-b*d^2*x^2+b*c^2)^(3/2)+81/64*(d*x+c)^(1/2)/b/c^4/(-b*d^2*x^2+b*c^2)^(3/2 
)-277/384*(d*x+c)^(3/2)/b/c^5/(-b*d^2*x^2+b*c^2)^(3/2)-21/256*(d*x+c)^(1/2 
)/b^2/c^6/(-b*d^2*x^2+b*c^2)^(1/2)-2*arctanh(b^(1/2)*c^(1/2)*(d*x+c)^(1/2) 
/(-b*d^2*x^2+b*c^2)^(1/2))/b^(5/2)/c^(13/2)+533/512*arctanh(2^(1/2)*b^(1/2 
)*c^(1/2)*(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2))*2^(1/2)/b^(5/2)/c^(13/2)
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (819 c^4+492 c^3 d x-690 c^2 d^2 x^2-428 c d^3 x^3+63 d^4 x^4\right )}{(c-d x)^2 (c+d x)^{7/2}}-3072 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )+1599 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )\right )}{1536 c^{13/2} \left (b \left (c^2-d^2 x^2\right )\right )^{5/2}} \] Input:

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

Output:

((c^2 - d^2*x^2)^(5/2)*((2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(819*c^4 + 492*c^3* 
d*x - 690*c^2*d^2*x^2 - 428*c*d^3*x^3 + 63*d^4*x^4))/((c - d*x)^2*(c + d*x 
)^(7/2)) - 3072*ArcTanh[(Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]] + 159 
9*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]]))/( 
1536*c^(13/2)*(b*(c^2 - d^2*x^2))^(5/2))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {586, 114, 27, 168, 27, 168, 27, 169, 27, 169, 27, 174, 73, 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.

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

\(\Big \downarrow \) 586

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\int \frac {3 b d (4 c-3 d x)}{2 x (c+d x)^3 (b c-b d x)^{5/2}}dx}{6 b c^2 d}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\int \frac {4 c-3 d x}{x (c+d x)^3 (b c-b d x)^{5/2}}dx}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\int \frac {b c d (32 c-49 d x)}{2 x (c+d x)^2 (b c-b d x)^{5/2}}dx}{4 b c^2 d}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\int \frac {32 c-49 d x}{x (c+d x)^2 (b c-b d x)^{5/2}}dx}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\int \frac {b c d (128 c-405 d x)}{2 x (c+d x) (b c-b d x)^{5/2}}dx}{2 b c^2 d}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\int \frac {128 c-405 d x}{x (c+d x) (b c-b d x)^{5/2}}dx}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {-\frac {\int -\frac {3 b c d (256 c-277 d x)}{2 x (c+d x) (b c-b d x)^{3/2}}dx}{3 b^2 c^2 d}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\frac {\int \frac {256 c-277 d x}{x (c+d x) (b c-b d x)^{3/2}}dx}{2 b c}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\frac {-\frac {\int -\frac {b c d (512 c-21 d x)}{2 x (c+d x) \sqrt {b c-b d x}}dx}{b^2 c^2 d}-\frac {21}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\frac {\frac {\int \frac {512 c-21 d x}{x (c+d x) \sqrt {b c-b d x}}dx}{2 b c}-\frac {21}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\frac {\frac {512 \int \frac {1}{x \sqrt {b c-b d x}}dx-533 d \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{2 b c}-\frac {21}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\frac {\frac {\frac {1066 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b}-\frac {1024 \int \frac {1}{\frac {c}{d}-\frac {b c-b d x}{b d}}d\sqrt {b c-b d x}}{b d}}{2 b c}-\frac {21}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\frac {\frac {\frac {\frac {\frac {533 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c}}-\frac {1024 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c}}}{2 b c}-\frac {21}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {277}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {81}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {7}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}}{4 c^2}+\frac {1}{6 b c^2 (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

Input:

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

Output:

(Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*(1/(6*b*c^2*(c + d*x)^3*(b*c - b*d*x)^(3/ 
2)) + (7/(4*b*c*(c + d*x)^2*(b*c - b*d*x)^(3/2)) + (81/(2*b*c*(c + d*x)*(b 
*c - b*d*x)^(3/2)) + (-277/(3*b*c*(b*c - b*d*x)^(3/2)) + (-21/(b*c*Sqrt[b* 
c - b*d*x]) + ((-1024*ArcTanh[Sqrt[b*c - b*d*x]/(Sqrt[b]*Sqrt[c])])/(Sqrt[ 
b]*Sqrt[c]) + (533*Sqrt[2]*ArcTanh[Sqrt[b*c - b*d*x]/(Sqrt[2]*Sqrt[b]*Sqrt 
[c])])/(Sqrt[b]*Sqrt[c]))/(2*b*c))/(2*b*c))/(4*c))/(8*c))/(4*c^2)))/Sqrt[b 
*c^2 - b*d^2*x^2]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 586

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( 
b*x)/d)^FracPart[p])   Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.35

method	result	size
default	\(\frac {\sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (-1599 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) d^{4} x^{4}-3198 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) c \,d^{3} x^{3}+3072 \sqrt {\left (-d x +c \right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) d^{4} x^{4}+6144 \sqrt {\left (-d x +c \right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) c \,d^{3} x^{3}+126 \sqrt {b c}\, d^{4} x^{4}+3198 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) c^{3} d x -856 \sqrt {b c}\, c \,d^{3} x^{3}+1599 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) c^{4}-6144 \sqrt {\left (-d x +c \right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) c^{3} d x -1380 \sqrt {b c}\, c^{2} d^{2} x^{2}-3072 \,\operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) c^{4} \sqrt {\left (-d x +c \right ) b}+984 \sqrt {b c}\, c^{3} d x +1638 \sqrt {b c}\, c^{4}\right )}{1536 b^{3} \left (d x +c \right )^{\frac {7}{2}} \left (-d x +c \right )^{2} c^{6} \sqrt {b c}}\)	\(413\)

Input:

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

Output:

1/1536*(b*(-d^2*x^2+c^2))^(1/2)/b^3*(-1599*((-d*x+c)*b)^(1/2)*2^(1/2)*arct 
anh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*d^4*x^4-3198*((-d*x+c)*b)^ 
(1/2)*2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*c*d^3*x^ 
3+3072*((-d*x+c)*b)^(1/2)*arctanh(((-d*x+c)*b)^(1/2)/(b*c)^(1/2))*d^4*x^4+ 
6144*((-d*x+c)*b)^(1/2)*arctanh(((-d*x+c)*b)^(1/2)/(b*c)^(1/2))*c*d^3*x^3+ 
126*(b*c)^(1/2)*d^4*x^4+3198*((-d*x+c)*b)^(1/2)*2^(1/2)*arctanh(1/2*((-d*x 
+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*c^3*d*x-856*(b*c)^(1/2)*c*d^3*x^3+1599*( 
(-d*x+c)*b)^(1/2)*2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/ 
2))*c^4-6144*((-d*x+c)*b)^(1/2)*arctanh(((-d*x+c)*b)^(1/2)/(b*c)^(1/2))*c^ 
3*d*x-1380*(b*c)^(1/2)*c^2*d^2*x^2-3072*arctanh(((-d*x+c)*b)^(1/2)/(b*c)^( 
1/2))*c^4*((-d*x+c)*b)^(1/2)+984*(b*c)^(1/2)*c^3*d*x+1638*(b*c)^(1/2)*c^4) 
/(d*x+c)^(7/2)/(-d*x+c)^2/c^6/(b*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 837, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\left [\frac {1599 \, \sqrt {2} {\left (d^{6} x^{6} + 2 \, c d^{5} x^{5} - c^{2} d^{4} x^{4} - 4 \, c^{3} d^{3} x^{3} - c^{4} d^{2} x^{2} + 2 \, c^{5} d x + c^{6}\right )} \sqrt {b c} \log \left (-\frac {b d^{2} x^{2} - 2 \, b c d x - 3 \, b c^{2} - 2 \, \sqrt {2} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {b c} \sqrt {d x + c}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 3072 \, {\left (d^{6} x^{6} + 2 \, c d^{5} x^{5} - c^{2} d^{4} x^{4} - 4 \, c^{3} d^{3} x^{3} - c^{4} d^{2} x^{2} + 2 \, c^{5} d x + c^{6}\right )} \sqrt {b c} \log \left (-\frac {b d^{2} x^{2} - b c d x - 2 \, b c^{2} + 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {b c} \sqrt {d x + c}}{d x^{2} + c x}\right ) + 4 \, {\left (63 \, c d^{4} x^{4} - 428 \, c^{2} d^{3} x^{3} - 690 \, c^{3} d^{2} x^{2} + 492 \, c^{4} d x + 819 \, c^{5}\right )} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c}}{3072 \, {\left (b^{3} c^{7} d^{6} x^{6} + 2 \, b^{3} c^{8} d^{5} x^{5} - b^{3} c^{9} d^{4} x^{4} - 4 \, b^{3} c^{10} d^{3} x^{3} - b^{3} c^{11} d^{2} x^{2} + 2 \, b^{3} c^{12} d x + b^{3} c^{13}\right )}}, -\frac {1599 \, \sqrt {2} {\left (d^{6} x^{6} + 2 \, c d^{5} x^{5} - c^{2} d^{4} x^{4} - 4 \, c^{3} d^{3} x^{3} - c^{4} d^{2} x^{2} + 2 \, c^{5} d x + c^{6}\right )} \sqrt {-b c} \arctan \left (\frac {\sqrt {2} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {-b c} \sqrt {d x + c}}{2 \, {\left (b c d x + b c^{2}\right )}}\right ) - 3072 \, {\left (d^{6} x^{6} + 2 \, c d^{5} x^{5} - c^{2} d^{4} x^{4} - 4 \, c^{3} d^{3} x^{3} - c^{4} d^{2} x^{2} + 2 \, c^{5} d x + c^{6}\right )} \sqrt {-b c} \arctan \left (\frac {\sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {-b c} \sqrt {d x + c}}{b c d x + b c^{2}}\right ) - 2 \, {\left (63 \, c d^{4} x^{4} - 428 \, c^{2} d^{3} x^{3} - 690 \, c^{3} d^{2} x^{2} + 492 \, c^{4} d x + 819 \, c^{5}\right )} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c}}{1536 \, {\left (b^{3} c^{7} d^{6} x^{6} + 2 \, b^{3} c^{8} d^{5} x^{5} - b^{3} c^{9} d^{4} x^{4} - 4 \, b^{3} c^{10} d^{3} x^{3} - b^{3} c^{11} d^{2} x^{2} + 2 \, b^{3} c^{12} d x + b^{3} c^{13}\right )}}\right ] \] Input:

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

Output:

[1/3072*(1599*sqrt(2)*(d^6*x^6 + 2*c*d^5*x^5 - c^2*d^4*x^4 - 4*c^3*d^3*x^3 
 - c^4*d^2*x^2 + 2*c^5*d*x + c^6)*sqrt(b*c)*log(-(b*d^2*x^2 - 2*b*c*d*x - 
3*b*c^2 - 2*sqrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(b*c)*sqrt(d*x + c))/(d^2 
*x^2 + 2*c*d*x + c^2)) + 3072*(d^6*x^6 + 2*c*d^5*x^5 - c^2*d^4*x^4 - 4*c^3 
*d^3*x^3 - c^4*d^2*x^2 + 2*c^5*d*x + c^6)*sqrt(b*c)*log(-(b*d^2*x^2 - b*c* 
d*x - 2*b*c^2 + 2*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(b*c)*sqrt(d*x + c))/(d*x^2 
 + c*x)) + 4*(63*c*d^4*x^4 - 428*c^2*d^3*x^3 - 690*c^3*d^2*x^2 + 492*c^4*d 
*x + 819*c^5)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c))/(b^3*c^7*d^6*x^6 + 2 
*b^3*c^8*d^5*x^5 - b^3*c^9*d^4*x^4 - 4*b^3*c^10*d^3*x^3 - b^3*c^11*d^2*x^2 
 + 2*b^3*c^12*d*x + b^3*c^13), -1/1536*(1599*sqrt(2)*(d^6*x^6 + 2*c*d^5*x^ 
5 - c^2*d^4*x^4 - 4*c^3*d^3*x^3 - c^4*d^2*x^2 + 2*c^5*d*x + c^6)*sqrt(-b*c 
)*arctan(1/2*sqrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(-b*c)*sqrt(d*x + c)/(b* 
c*d*x + b*c^2)) - 3072*(d^6*x^6 + 2*c*d^5*x^5 - c^2*d^4*x^4 - 4*c^3*d^3*x^ 
3 - c^4*d^2*x^2 + 2*c^5*d*x + c^6)*sqrt(-b*c)*arctan(sqrt(-b*d^2*x^2 + b*c 
^2)*sqrt(-b*c)*sqrt(d*x + c)/(b*c*d*x + b*c^2)) - 2*(63*c*d^4*x^4 - 428*c^ 
2*d^3*x^3 - 690*c^3*d^2*x^2 + 492*c^4*d*x + 819*c^5)*sqrt(-b*d^2*x^2 + b*c 
^2)*sqrt(d*x + c))/(b^3*c^7*d^6*x^6 + 2*b^3*c^8*d^5*x^5 - b^3*c^9*d^4*x^4 
- 4*b^3*c^10*d^3*x^3 - b^3*c^11*d^2*x^2 + 2*b^3*c^12*d*x + b^3*c^13)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=-\frac {533 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{512 \, \sqrt {-b c} b^{2} c^{6}} + \frac {2 \, \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{\sqrt {-b c}}\right )}{\sqrt {-b c} b^{2} c^{6}} + \frac {256 \, b^{4} c^{4} - 1920 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )} b^{3} c^{3} - 1596 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )}^{2} b^{2} c^{2} - 176 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )}^{3} b c + 63 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )}^{4}}{768 \, {\left (2 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c} b c - {\left (-{\left (d x + c\right )} b + 2 \, b c\right )}^{\frac {3}{2}}\right )}^{3} b^{2} c^{6}} \] Input:

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

Output:

-533/512*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-(d*x + c)*b + 2*b*c)/sqrt(-b*c)) 
/(sqrt(-b*c)*b^2*c^6) + 2*arctan(sqrt(-(d*x + c)*b + 2*b*c)/sqrt(-b*c))/(s 
qrt(-b*c)*b^2*c^6) + 1/768*(256*b^4*c^4 - 1920*((d*x + c)*b - 2*b*c)*b^3*c 
^3 - 1596*((d*x + c)*b - 2*b*c)^2*b^2*c^2 - 176*((d*x + c)*b - 2*b*c)^3*b* 
c + 63*((d*x + c)*b - 2*b*c)^4)/((2*sqrt(-(d*x + c)*b + 2*b*c)*b*c - (-(d* 
x + c)*b + 2*b*c)^(3/2))^3*b^2*c^6)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {b}\, \left (-1599 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{4}-3198 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{3} d x +3198 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c \,d^{3} x^{3}+1599 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) d^{4} x^{4}+1599 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{4}+3198 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{3} d x -3198 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c \,d^{3} x^{3}-1599 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) d^{4} x^{4}+3072 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) c^{4}+6144 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) c^{3} d x -6144 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) c \,d^{3} x^{3}-3072 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) d^{4} x^{4}-3072 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) c^{4}-6144 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) c^{3} d x +6144 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) c \,d^{3} x^{3}+3072 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) d^{4} x^{4}+3276 c^{5}+1968 c^{4} d x -2760 c^{3} d^{2} x^{2}-1712 c^{2} d^{3} x^{3}+252 c \,d^{4} x^{4}\right )}{3072 \sqrt {-d x +c}\, b^{3} c^{7} \left (-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}\right )} \] Input:

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

Output:

(sqrt(b)*( - 1599*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c 
)*sqrt(2))*c**4 - 3198*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - s 
qrt(c)*sqrt(2))*c**3*d*x + 3198*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - 
 d*x) - sqrt(c)*sqrt(2))*c*d**3*x**3 + 1599*sqrt(c)*sqrt(c - d*x)*sqrt(2)* 
log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*d**4*x**4 + 1599*sqrt(c)*sqrt(c - d*x 
)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**4 + 3198*sqrt(c)*sqrt(c 
- d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**3*d*x - 3198*sqrt(c 
)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c*d**3*x**3 - 
 1599*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*d 
**4*x**4 + 3072*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c))*c**4 + 
6144*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c))*c**3*d*x - 6144*sq 
rt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c))*c*d**3*x**3 - 3072*sqrt(c 
)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c))*d**4*x**4 - 3072*sqrt(c)*sqrt 
(c - d*x)*log(sqrt(c - d*x) + sqrt(c))*c**4 - 6144*sqrt(c)*sqrt(c - d*x)*l 
og(sqrt(c - d*x) + sqrt(c))*c**3*d*x + 6144*sqrt(c)*sqrt(c - d*x)*log(sqrt 
(c - d*x) + sqrt(c))*c*d**3*x**3 + 3072*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - 
 d*x) + sqrt(c))*d**4*x**4 + 3276*c**5 + 1968*c**4*d*x - 2760*c**3*d**2*x* 
*2 - 1712*c**2*d**3*x**3 + 252*c*d**4*x**4))/(3072*sqrt(c - d*x)*b**3*c**7 
*(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 - d**4*x**4))