\(\int \frac {1}{x^2 (c+d x)^{3/2} (b c^2-b d^2 x^2)^{5/2}} \, dx\) [841]

Optimal result

Integrand size = 32, antiderivative size = 353 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=-\frac {7 d}{6 b c^3 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}-\frac {1}{b c^2 x (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{3/2}}-\frac {27 d}{16 b c^4 \sqrt {c+d x} \left (b c^2-b d^2 x^2\right )^{3/2}}-\frac {237 d \sqrt {c+d x}}{64 b c^5 \left (b c^2-b d^2 x^2\right )^{3/2}}+\frac {331 d (c+d x)^{3/2}}{128 b c^6 \left (b c^2-b d^2 x^2\right )^{3/2}}+\frac {609 d \sqrt {c+d x}}{256 b^2 c^7 \sqrt {b c^2-b d^2 x^2}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{b^{5/2} c^{15/2}}-\frac {1377 d \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{256 \sqrt {2} b^{5/2} c^{15/2}} \] Output:

-7/6*d/b/c^3/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(3/2)-1/b/c^2/x/(d*x+c)^(3/2 
)/(-b*d^2*x^2+b*c^2)^(3/2)-27/16*d/b/c^4/(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^ 
(3/2)-237/64*d*(d*x+c)^(1/2)/b/c^5/(-b*d^2*x^2+b*c^2)^(3/2)+331/128*d*(d*x 
+c)^(3/2)/b/c^6/(-b*d^2*x^2+b*c^2)^(3/2)+609/256*d*(d*x+c)^(1/2)/b^2/c^7/( 
-b*d^2*x^2+b*c^2)^(1/2)+3*d*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^(15/2)-1377/512*d*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^(15/2)
 

Mathematica [A] (verified)

Time = 1.87 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (-\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (768 c^5+1223 c^4 d x-2628 c^3 d^2 x^2-3114 c^2 d^3 x^3+1668 c d^4 x^4+1827 d^5 x^5\right )}{x (c-d x)^2 (c+d x)^{7/2}}+4608 d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )-4131 \sqrt {2} d \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )\right )}{1536 c^{15/2} \left (b \left (c^2-d^2 x^2\right )\right )^{5/2}} \] Input:

Integrate[1/(x^2*(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]*(768*c^5 + 1223*c^ 
4*d*x - 2628*c^3*d^2*x^2 - 3114*c^2*d^3*x^3 + 1668*c*d^4*x^4 + 1827*d^5*x^ 
5))/(x*(c - d*x)^2*(c + d*x)^(7/2)) + 4608*d*ArcTanh[(Sqrt[c]*Sqrt[c + d*x 
])/Sqrt[c^2 - d^2*x^2]] - 4131*Sqrt[2]*d*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + 
 d*x])/Sqrt[c^2 - d^2*x^2]]))/(1536*c^(15/2)*(b*(c^2 - d^2*x^2))^(5/2))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.98, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {586, 114, 27, 168, 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.

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

\(\Big \downarrow \) 586

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {\int \frac {b d (3 c-11 d x)}{2 x (c+d x)^4 (b c-b d x)^{5/2}}dx}{b c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \int \frac {3 c-11 d x}{x (c+d x)^4 (b c-b d x)^{5/2}}dx}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {\int \frac {9 b c d (2 c-7 d x)}{x (c+d x)^3 (b c-b d x)^{5/2}}dx}{6 b c^2 d}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\int \frac {b c d (16 c-63 d x)}{2 x (c+d x)^2 (b c-b d x)^{5/2}}dx}{4 b c^2 d}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\int \frac {16 c-63 d x}{x (c+d x)^2 (b c-b d x)^{5/2}}dx}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\int \frac {b c d (64 c-395 d x)}{2 x (c+d x) (b c-b d x)^{5/2}}dx}{2 b c^2 d}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\int \frac {64 c-395 d x}{x (c+d x) (b c-b d x)^{5/2}}dx}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {-\frac {\int -\frac {3 b c d (128 c-331 d x)}{2 x (c+d x) (b c-b d x)^{3/2}}dx}{3 b^2 c^2 d}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\frac {\int \frac {128 c-331 d x}{x (c+d x) (b c-b d x)^{3/2}}dx}{2 b c}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\frac {-\frac {\int -\frac {b c d (256 c-203 d x)}{2 x (c+d x) \sqrt {b c-b d x}}dx}{b^2 c^2 d}-\frac {203}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\frac {\frac {\int \frac {256 c-203 d x}{x (c+d x) \sqrt {b c-b d x}}dx}{2 b c}-\frac {203}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\frac {\frac {256 \int \frac {1}{x \sqrt {b c-b d x}}dx-459 d \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{2 b c}-\frac {203}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\frac {\frac {\frac {918 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b}-\frac {512 \int \frac {1}{\frac {c}{d}-\frac {b c-b d x}{b d}}d\sqrt {b c-b d x}}{b d}}{2 b c}-\frac {203}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (-\frac {d \left (\frac {3 \left (\frac {\frac {\frac {\frac {\frac {459 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c}}-\frac {512 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c}}}{2 b c}-\frac {203}{b c \sqrt {b c-b d x}}}{2 b c}-\frac {331}{3 b c (b c-b d x)^{3/2}}}{4 c}+\frac {79}{2 b c (c+d x) (b c-b d x)^{3/2}}}{8 c}+\frac {9}{4 b c (c+d x)^2 (b c-b d x)^{3/2}}\right )}{2 c}+\frac {7}{3 b c (c+d x)^3 (b c-b d x)^{3/2}}\right )}{2 c^2}-\frac {1}{b c^2 x (c+d x)^3 (b c-b d x)^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

Input:

Int[1/(x^2*(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/(b*c^2*x*(c + d*x)^3*(b*c - b*d*x)^( 
3/2))) - (d*(7/(3*b*c*(c + d*x)^3*(b*c - b*d*x)^(3/2)) + (3*(9/(4*b*c*(c + 
 d*x)^2*(b*c - b*d*x)^(3/2)) + (79/(2*b*c*(c + d*x)*(b*c - b*d*x)^(3/2)) + 
 (-331/(3*b*c*(b*c - b*d*x)^(3/2)) + (-203/(b*c*Sqrt[b*c - b*d*x]) + ((-51 
2*ArcTanh[Sqrt[b*c - b*d*x]/(Sqrt[b]*Sqrt[c])])/(Sqrt[b]*Sqrt[c]) + (459*S 
qrt[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)))/(2*c)))/(2*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 = 0.32 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {\left (-d x +c \right ) \sqrt {-\frac {b \left (d^{2} x^{2}-c^{2}\right )}{d x +c}}\, \sqrt {d x +c}}{c^{7} x \sqrt {-b \left (d x -c \right )}\, \sqrt {-b \left (d^{2} x^{2}-c^{2}\right )}\, b^{2}}+\frac {d \left (\frac {1}{\sqrt {-b d x +b c}}+\frac {b c}{12 \left (-b d x +b c \right )^{\frac {3}{2}}}+\frac {\frac {225 \left (-b d x +b c \right )^{\frac {5}{2}}}{32}-\frac {179 b c \left (-b d x +b c \right )^{\frac {3}{2}}}{6}+\frac {255 b^{2} c^{2} \sqrt {-b d x +b c}}{8}}{4 \left (-b d x -b c \right )^{3}}-\frac {1377 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b d x +b c}\, \sqrt {2}}{2 \sqrt {b c}}\right )}{256 \sqrt {b c}}+\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {-b d x +b c}}{\sqrt {b c}}\right )}{\sqrt {b c}}\right ) \sqrt {-\frac {b \left (d^{2} x^{2}-c^{2}\right )}{d x +c}}\, \sqrt {d x +c}}{2 c^{7} \sqrt {-b \left (d^{2} x^{2}-c^{2}\right )}\, b^{2}}\)	\(286\)
default	\(-\frac {\sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (-4131 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) d^{5} x^{5}-8262 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) c \,d^{4} x^{4}+4608 \sqrt {\left (-d x +c \right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) d^{5} x^{5}+9216 \sqrt {\left (-d x +c \right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) c \,d^{4} x^{4}+3654 \sqrt {b c}\, d^{5} x^{5}+8262 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) c^{3} d^{2} x^{2}+3336 \sqrt {b c}\, c \,d^{4} x^{4}+4131 \sqrt {\left (-d x +c \right ) b}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) c^{4} d x -9216 \sqrt {\left (-d x +c \right ) b}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) c^{3} d^{2} x^{2}-6228 \sqrt {b c}\, c^{2} d^{3} x^{3}-4608 \,\operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right ) c^{4} d x \sqrt {\left (-d x +c \right ) b}-5256 \sqrt {b c}\, c^{3} d^{2} x^{2}+2446 \sqrt {b c}\, c^{4} d x +1536 \sqrt {b c}\, c^{5}\right )}{1536 b^{3} \left (d x +c \right )^{\frac {7}{2}} \left (-d x +c \right )^{2} c^{7} \sqrt {b c}\, x}\)	\(444\)

Input:

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

Output:

-(-d*x+c)/c^7/x/(-b*(d*x-c))^(1/2)*(-1/(d*x+c)*b*(d^2*x^2-c^2))^(1/2)*(d*x 
+c)^(1/2)/(-b*(d^2*x^2-c^2))^(1/2)/b^2+1/2/c^7*d*(1/(-b*d*x+b*c)^(1/2)+1/1 
2*b*c/(-b*d*x+b*c)^(3/2)+1/4*(225/32*(-b*d*x+b*c)^(5/2)-179/6*b*c*(-b*d*x+ 
b*c)^(3/2)+255/8*b^2*c^2*(-b*d*x+b*c)^(1/2))/(-b*d*x-b*c)^3-1377/256*2^(1/ 
2)/(b*c)^(1/2)*arctanh(1/2*(-b*d*x+b*c)^(1/2)*2^(1/2)/(b*c)^(1/2))+6/(b*c) 
^(1/2)*arctanh((-b*d*x+b*c)^(1/2)/(b*c)^(1/2)))*(-1/(d*x+c)*b*(d^2*x^2-c^2 
))^(1/2)*(d*x+c)^(1/2)/(-b*(d^2*x^2-c^2))^(1/2)/b^2
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.53 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/3072*(4131*sqrt(2)*(d^7*x^7 + 2*c*d^6*x^6 - c^2*d^5*x^5 - 4*c^3*d^4*x^4 
 - c^4*d^3*x^3 + 2*c^5*d^2*x^2 + c^6*d*x)*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)) + 4608*(d^7*x^7 + 2*c*d^6*x^6 - c^2*d^5*x^5 
 - 4*c^3*d^4*x^4 - c^4*d^3*x^3 + 2*c^5*d^2*x^2 + c^6*d*x)*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*(1827*c*d^5*x^5 + 1668*c^2*d^4*x^4 - 3114*c^3 
*d^3*x^3 - 2628*c^4*d^2*x^2 + 1223*c^5*d*x + 768*c^6)*sqrt(-b*d^2*x^2 + b* 
c^2)*sqrt(d*x + c))/(b^3*c^8*d^6*x^7 + 2*b^3*c^9*d^5*x^6 - b^3*c^10*d^4*x^ 
5 - 4*b^3*c^11*d^3*x^4 - b^3*c^12*d^2*x^3 + 2*b^3*c^13*d*x^2 + b^3*c^14*x) 
, 1/1536*(4131*sqrt(2)*(d^7*x^7 + 2*c*d^6*x^6 - c^2*d^5*x^5 - 4*c^3*d^4*x^ 
4 - c^4*d^3*x^3 + 2*c^5*d^2*x^2 + c^6*d*x)*sqrt(-b*c)*arctan(1/2*sqrt(2)*s 
qrt(-b*d^2*x^2 + b*c^2)*sqrt(-b*c)*sqrt(d*x + c)/(b*c*d*x + b*c^2)) - 4608 
*(d^7*x^7 + 2*c*d^6*x^6 - c^2*d^5*x^5 - 4*c^3*d^4*x^4 - c^4*d^3*x^3 + 2*c^ 
5*d^2*x^2 + c^6*d*x)*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*(1827*c*d^5*x^5 + 1668*c^2*d^4*x^4 - 
 3114*c^3*d^3*x^3 - 2628*c^4*d^2*x^2 + 1223*c^5*d*x + 768*c^6)*sqrt(-b*d^2 
*x^2 + b*c^2)*sqrt(d*x + c))/(b^3*c^8*d^6*x^7 + 2*b^3*c^9*d^5*x^6 - b^3*c^ 
10*d^4*x^5 - 4*b^3*c^11*d^3*x^4 - b^3*c^12*d^2*x^3 + 2*b^3*c^13*d*x^2 + b^ 
3*c^14*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\frac {1}{1536} \, d {\left (\frac {4131 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{\sqrt {-b c} b^{2} c^{7}} - \frac {4608 \, \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{\sqrt {-b c}}\right )}{\sqrt {-b c} b^{2} c^{7}} - \frac {1536 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{{\left ({\left (d x + c\right )} b - b c\right )} b^{2} c^{7}} + \frac {2 \, {\left (256 \, b^{4} c^{4} - 2688 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )} b^{3} c^{3} - 7476 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )}^{2} b^{2} c^{2} - 5136 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )}^{3} b c - 1059 \, {\left ({\left (d x + c\right )} b - 2 \, b c\right )}^{4}\right )}}{{\left (2 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c} b c - {\left (-{\left (d x + c\right )} b + 2 \, b c\right )}^{\frac {3}{2}}\right )}^{3} b^{2} c^{7}}\right )} \] Input:

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

Output:

1/1536*d*(4131*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-(d*x + c)*b + 2*b*c)/sqrt( 
-b*c))/(sqrt(-b*c)*b^2*c^7) - 4608*arctan(sqrt(-(d*x + c)*b + 2*b*c)/sqrt( 
-b*c))/(sqrt(-b*c)*b^2*c^7) - 1536*sqrt(-(d*x + c)*b + 2*b*c)/(((d*x + c)* 
b - b*c)*b^2*c^7) + 2*(256*b^4*c^4 - 2688*((d*x + c)*b - 2*b*c)*b^3*c^3 - 
7476*((d*x + c)*b - 2*b*c)^2*b^2*c^2 - 5136*((d*x + c)*b - 2*b*c)^3*b*c - 
1059*((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^7))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \left (b c^2-b d^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {b}\, \left (4131 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{4} d x +8262 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{3} d^{2} x^{2}-8262 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c \,d^{4} x^{4}-4131 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) d^{5} x^{5}-4131 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{4} d x -8262 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{3} d^{2} x^{2}+8262 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c \,d^{4} x^{4}+4131 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) d^{5} x^{5}-4608 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) c^{4} d x -9216 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) c^{3} d^{2} x^{2}+9216 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) c \,d^{4} x^{4}+4608 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right ) d^{5} x^{5}+4608 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) c^{4} d x +9216 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) c^{3} d^{2} x^{2}-9216 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) c \,d^{4} x^{4}-4608 \sqrt {c}\, \sqrt {-d x +c}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right ) d^{5} x^{5}-3072 c^{6}-4892 c^{5} d x +10512 c^{4} d^{2} x^{2}+12456 c^{3} d^{3} x^{3}-6672 c^{2} d^{4} x^{4}-7308 c \,d^{5} x^{5}\right )}{3072 \sqrt {-d x +c}\, b^{3} c^{8} x \left (-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}\right )} \] Input:

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

Output:

(sqrt(b)*(4131*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*s 
qrt(2))*c**4*d*x + 8262*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - 
sqrt(c)*sqrt(2))*c**3*d**2*x**2 - 8262*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(s 
qrt(c - d*x) - sqrt(c)*sqrt(2))*c*d**4*x**4 - 4131*sqrt(c)*sqrt(c - d*x)*s 
qrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*d**5*x**5 - 4131*sqrt(c)*sqrt( 
c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**4*d*x - 8262*sqrt 
(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**3*d**2*x 
**2 + 8262*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt( 
2))*c*d**4*x**4 + 4131*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + s 
qrt(c)*sqrt(2))*d**5*x**5 - 4608*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - 
 sqrt(c))*c**4*d*x - 9216*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c 
))*c**3*d**2*x**2 + 9216*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c) 
)*c*d**4*x**4 + 4608*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) - sqrt(c))*d* 
*5*x**5 + 4608*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) + sqrt(c))*c**4*d*x 
 + 9216*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) + sqrt(c))*c**3*d**2*x**2 
- 9216*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) + sqrt(c))*c*d**4*x**4 - 46 
08*sqrt(c)*sqrt(c - d*x)*log(sqrt(c - d*x) + sqrt(c))*d**5*x**5 - 3072*c** 
6 - 4892*c**5*d*x + 10512*c**4*d**2*x**2 + 12456*c**3*d**3*x**3 - 6672*c** 
2*d**4*x**4 - 7308*c*d**5*x**5))/(3072*sqrt(c - d*x)*b**3*c**8*x*(c**4 + 2 
*c**3*d*x - 2*c*d**3*x**3 - d**4*x**4))