\(\int \frac {(c^2-d^2 x^2)^{3/2}}{(e x)^{3/2} (c+d x)^{5/2}} \, dx\) [979]

Optimal result

Integrand size = 33, antiderivative size = 160 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=-\frac {2 \sqrt {c^2-d^2 x^2}}{e \sqrt {e x} \sqrt {c+d x}}-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {e} \sqrt {c^2-d^2 x^2}}{\sqrt {d} \sqrt {e x} \sqrt {c+d x}}\right )}{e^{3/2}}+\frac {4 \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {e} \sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}\right )}{e^{3/2}} \] Output:

-2*(-d^2*x^2+c^2)^(1/2)/e/(e*x)^(1/2)/(d*x+c)^(1/2)-2*d^(1/2)*arctan(e^(1/ 
2)*(-d^2*x^2+c^2)^(1/2)/d^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2))/e^(3/2)+4*2^(1/ 
2)*d^(1/2)*arctan(1/2*e^(1/2)*(-d^2*x^2+c^2)^(1/2)*2^(1/2)/d^(1/2)/(e*x)^( 
1/2)/(d*x+c)^(1/2))/e^(3/2)
 

Mathematica [A] (verified)

Time = 9.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=-\frac {2 x \sqrt {c^2-d^2 x^2} \left (-\sqrt {d} \sqrt {x} \sqrt {c-d x} \arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )+\sqrt {c} \sqrt {1-\frac {d x}{c}} \left (\sqrt {c-d x}+2 \sqrt {2} \sqrt {d} \sqrt {x} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {x}}{\sqrt {c-d x}}\right )\right )\right )}{\sqrt {c} (e x)^{3/2} \sqrt {c-d x} \sqrt {c+d x} \sqrt {1-\frac {d x}{c}}} \] Input:

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

Output:

(-2*x*Sqrt[c^2 - d^2*x^2]*(-(Sqrt[d]*Sqrt[x]*Sqrt[c - d*x]*ArcSin[(Sqrt[d] 
*Sqrt[x])/Sqrt[c]]) + Sqrt[c]*Sqrt[1 - (d*x)/c]*(Sqrt[c - d*x] + 2*Sqrt[2] 
*Sqrt[d]*Sqrt[x]*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[x])/Sqrt[c - d*x]])))/(Sqrt[ 
c]*(e*x)^(3/2)*Sqrt[c - d*x]*Sqrt[c + d*x]*Sqrt[1 - (d*x)/c])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {586, 109, 27, 175, 65, 104, 218}

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.

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

\(\Big \downarrow \) 586

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \int \frac {(c-d x)^{3/2}}{(e x)^{3/2} (c+d x)}dx}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (-\frac {2 \int \frac {c d e (3 c-d x)}{2 \sqrt {e x} \sqrt {c-d x} (c+d x)}dx}{c e^2}-\frac {2 \sqrt {c-d x}}{e \sqrt {e x}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (-\frac {d \int \frac {3 c-d x}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx}{e}-\frac {2 \sqrt {c-d x}}{e \sqrt {e x}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (-\frac {d \left (4 c \int \frac {1}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx-\int \frac {1}{\sqrt {e x} \sqrt {c-d x}}dx\right )}{e}-\frac {2 \sqrt {c-d x}}{e \sqrt {e x}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (-\frac {d \left (4 c \int \frac {1}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx-2 \int \frac {1}{\frac {d x e}{c-d x}+e}d\frac {\sqrt {e x}}{\sqrt {c-d x}}\right )}{e}-\frac {2 \sqrt {c-d x}}{e \sqrt {e x}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (-\frac {d \left (8 c \int \frac {1}{c e+\frac {2 c d x e}{c-d x}}d\frac {\sqrt {e x}}{\sqrt {c-d x}}-2 \int \frac {1}{\frac {d x e}{c-d x}+e}d\frac {\sqrt {e x}}{\sqrt {c-d x}}\right )}{e}-\frac {2 \sqrt {c-d x}}{e \sqrt {e x}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (-\frac {d \left (\frac {4 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c-d x}}\right )}{\sqrt {d} \sqrt {e}}-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c-d x}}\right )}{\sqrt {d} \sqrt {e}}\right )}{e}-\frac {2 \sqrt {c-d x}}{e \sqrt {e x}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*((-2*Sqrt[c - d*x])/(e*Sqrt[e*x]) - (d*((-2*ArcTan[(S 
qrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c - d*x])])/(Sqrt[d]*Sqrt[e]) + (4*Sqrt[2] 
*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c - d*x])])/(Sqrt[d]*Sqr 
t[e])))/e))/(Sqrt[c - d*x]*Sqrt[c + d*x])
 

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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

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

rule 175

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

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.26

method	result	size
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) e x}}\right ) d e x \sqrt {-\frac {e \,c^{2}}{d}}-4 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c e x \sqrt {d e}+2 \sqrt {\left (-d x +c \right ) e x}\, \sqrt {d e}\, \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\right ) \sqrt {2}}{2 e \sqrt {e x}\, \sqrt {d x +c}\, \sqrt {\left (-d x +c \right ) e x}\, \sqrt {d e}\, \sqrt {-\frac {e \,c^{2}}{d}}}\)	\(201\)

Input:

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

Output:

-1/2*(-d^2*x^2+c^2)^(1/2)/e*(2^(1/2)*arctan(1/2*(d*e)^(1/2)*(-2*d*x+c)/d/( 
(-d*x+c)*e*x)^(1/2))*d*e*x*(-1/d*e*c^2)^(1/2)-4*ln((3*c*d*x*e+2*2^(1/2)*(- 
1/d*e*c^2)^(1/2)*((-d*x+c)*e*x)^(1/2)*d-e*c^2)/(d*x+c))*c*e*x*(d*e)^(1/2)+ 
2*((-d*x+c)*e*x)^(1/2)*(d*e)^(1/2)*2^(1/2)*(-1/d*e*c^2)^(1/2))*2^(1/2)/(e* 
x)^(1/2)/(d*x+c)^(1/2)/((-d*x+c)*e*x)^(1/2)/(d*e)^(1/2)/(-1/d*e*c^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.05 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\left [\frac {2 \, \sqrt {2} {\left (d e x^{2} + c e x\right )} \sqrt {-\frac {d}{e}} \log \left (-\frac {17 \, d^{3} x^{3} + 3 \, c d^{2} x^{2} - 13 \, c^{2} d x - 4 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} {\left (3 \, d x - c\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {d}{e}} + c^{3}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (d e x^{2} + c e x\right )} \sqrt {-\frac {d}{e}} \log \left (-\frac {8 \, d^{3} x^{3} - 7 \, c^{2} d x + c^{3} + 4 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (2 \, d x - c\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {d}{e}}}{d x + c}\right ) - 4 \, \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {e x}}{2 \, {\left (d e^{2} x^{2} + c e^{2} x\right )}}, \frac {2 \, \sqrt {2} {\left (d e x^{2} + c e x\right )} \sqrt {\frac {d}{e}} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} {\left (3 \, d x - c\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {\frac {d}{e}}}{4 \, {\left (d^{3} x^{3} - c^{2} d x\right )}}\right ) - {\left (d e x^{2} + c e x\right )} \sqrt {\frac {d}{e}} \arctan \left (\frac {\sqrt {-d^{2} x^{2} + c^{2}} {\left (2 \, d x - c\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {\frac {d}{e}}}{2 \, {\left (d^{3} x^{3} - c^{2} d x\right )}}\right ) - 2 \, \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {e x}}{d e^{2} x^{2} + c e^{2} x}\right ] \] Input:

integrate((-d^2*x^2+c^2)^(3/2)/(e*x)^(3/2)/(d*x+c)^(5/2),x, algorithm="fri 
cas")
 

Output:

[1/2*(2*sqrt(2)*(d*e*x^2 + c*e*x)*sqrt(-d/e)*log(-(17*d^3*x^3 + 3*c*d^2*x^ 
2 - 13*c^2*d*x - 4*sqrt(2)*sqrt(-d^2*x^2 + c^2)*(3*d*x - c)*sqrt(d*x + c)* 
sqrt(e*x)*sqrt(-d/e) + c^3)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + ( 
d*e*x^2 + c*e*x)*sqrt(-d/e)*log(-(8*d^3*x^3 - 7*c^2*d*x + c^3 + 4*sqrt(-d^ 
2*x^2 + c^2)*(2*d*x - c)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-d/e))/(d*x + c)) - 
4*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x))/(d*e^2*x^2 + c*e^2*x), (2* 
sqrt(2)*(d*e*x^2 + c*e*x)*sqrt(d/e)*arctan(1/4*sqrt(2)*sqrt(-d^2*x^2 + c^2 
)*(3*d*x - c)*sqrt(d*x + c)*sqrt(e*x)*sqrt(d/e)/(d^3*x^3 - c^2*d*x)) - (d* 
e*x^2 + c*e*x)*sqrt(d/e)*arctan(1/2*sqrt(-d^2*x^2 + c^2)*(2*d*x - c)*sqrt( 
d*x + c)*sqrt(e*x)*sqrt(d/e)/(d^3*x^3 - c^2*d*x)) - 2*sqrt(-d^2*x^2 + c^2) 
*sqrt(d*x + c)*sqrt(e*x))/(d*e^2*x^2 + c*e^2*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-d^2*x^2+c^2)^(3/2)/(e*x)^(3/2)/(d*x+c)^(5/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-d^2*x^2+c^2)^(3/2)/(e*x)^(3/2)/(d*x+c)^(5/2),x, algorithm="gia 
c")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {e}\, \left (-2 \sqrt {x}\, \sqrt {d}\, \sqrt {2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right )}{2}\right )}{\sqrt {2}+1}\right )+\sqrt {x}\, \sqrt {d}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {d}\, \sqrt {-d x +c}}{-d x +c}\right )-\sqrt {-d x +c}+\sqrt {x}\, \sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right )}{2}\right )+i \right ) i -\sqrt {x}\, \sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right )}{2}\right )-i \right ) i \right )}{\sqrt {x}\, e^{2}} \] Input:

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

Output:

(2*sqrt(e)*( - 2*sqrt(x)*sqrt(d)*sqrt(2)*atan(tan(asin((sqrt(x)*sqrt(d))/s 
qrt(c))/2)/(sqrt(2) + 1)) + sqrt(x)*sqrt(d)*atan((sqrt(x)*sqrt(d)*sqrt(c - 
 d*x))/(c - d*x)) - sqrt(c - d*x) + sqrt(x)*sqrt(d)*sqrt(2)*log( - sqrt(2) 
*i + tan(asin((sqrt(x)*sqrt(d))/sqrt(c))/2) + i)*i - sqrt(x)*sqrt(d)*sqrt( 
2)*log(sqrt(2)*i + tan(asin((sqrt(x)*sqrt(d))/sqrt(c))/2) - i)*i))/(sqrt(x 
)*e**2)