\(\int \frac {1}{(d+e x)^{3/2} (c d^2-c e^2 x^2)^{5/2}} \, dx\) [203]

Optimal result

Integrand size = 29, antiderivative size = 203 \[ \int \frac {1}{(d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{6 c d e (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}+\frac {d+7 e x}{32 c d^3 e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}+\frac {35 (d+3 e x)}{256 c^2 d^5 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}-\frac {105 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}}\right )}{256 \sqrt {2} c^{5/2} d^{11/2} e} \] Output:

-1/6/c/d/e/(e*x+d)^(3/2)/(-c*e^2*x^2+c*d^2)^(3/2)+1/32*(7*e*x+d)/c/d^3/e/( 
e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(3/2)+35/256*(3*e*x+d)/c^2/d^5/e/(e*x+d)^( 
1/2)/(-c*e^2*x^2+c*d^2)^(1/2)-105/512*arctanh(2^(1/2)*c^(1/2)*d^(1/2)*(e*x 
+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2))*2^(1/2)/c^(5/2)/d^(11/2)/e
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d} \left (d^4+612 d^3 e x+378 d^2 e^2 x^2-420 d e^3 x^3-315 e^4 x^4\right )-315 \sqrt {2} (d-e x) (d+e x)^{5/2} \sqrt {d^2-e^2 x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{1536 c^2 d^{11/2} e (d-e x) (d+e x)^{5/2} \sqrt {c \left (d^2-e^2 x^2\right )}} \] Input:

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

Output:

(2*Sqrt[d]*(d^4 + 612*d^3*e*x + 378*d^2*e^2*x^2 - 420*d*e^3*x^3 - 315*e^4* 
x^4) - 315*Sqrt[2]*(d - e*x)*(d + e*x)^(5/2)*Sqrt[d^2 - e^2*x^2]*ArcTanh[( 
Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[d^2 - e^2*x^2]])/(1536*c^2*d^(11/2)*e* 
(d - e*x)*(d + e*x)^(5/2)*Sqrt[c*(d^2 - e^2*x^2)])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.49, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {470, 470, 467, 470, 467, 471, 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.

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

\(\Big \downarrow \) 470

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

\(\Big \downarrow \) 470

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

\(\Big \downarrow \) 467

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

\(\Big \downarrow \) 470

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

\(\Big \downarrow \) 467

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {d+e x}}{c d e \sqrt {c d^2-c e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} c^{3/2} d^{3/2} e}\right )}{4 d}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}\right )}{6 c d}+\frac {\sqrt {d+e x}}{3 c d e \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{4 d}-\frac {1}{6 c d e (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

Input:

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

Output:

-1/6*1/(c*d*e*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(3/2)) + (3*(-1/4*1/(c*d 
*e*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2)) + (7*(Sqrt[d + e*x]/(3*c*d*e*( 
c*d^2 - c*e^2*x^2)^(3/2)) + (5*(-1/2*1/(c*d*e*Sqrt[d + e*x]*Sqrt[c*d^2 - c 
*e^2*x^2]) + (3*(Sqrt[d + e*x]/(c*d*e*Sqrt[c*d^2 - c*e^2*x^2]) - ArcTanh[S 
qrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])]/(Sqrt[2]*c 
^(3/2)*d^(3/2)*e)))/(4*d)))/(6*c*d)))/(8*d)))/(4*d)
 

Defintions of rubi rules used

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 467

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-c)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*a*d*(p + 1))), x] + Simp[c*((n + 2 
*p + 2)/(2*a*(p + 1)))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[p, -1] && LtQ[0, 
n, 1] && IntegerQ[2*p]
 

rule 470

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[(n + 
2*p + 2)/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + 
 p + 1, 0] && IntegerQ[2*p]
 

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.37

method	result	size
default	\(-\frac {\sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-315 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, e^{4} x^{4} \sqrt {c \left (-e x +d \right )}-630 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, d \,e^{3} x^{3} \sqrt {c \left (-e x +d \right )}+630 \sqrt {c d}\, e^{4} x^{4}+630 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, d^{3} e x \sqrt {c \left (-e x +d \right )}+840 \sqrt {c d}\, d \,e^{3} x^{3}+315 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, d^{4} \sqrt {c \left (-e x +d \right )}-756 \sqrt {c d}\, d^{2} e^{2} x^{2}-1224 \sqrt {c d}\, d^{3} e x -2 \sqrt {c d}\, d^{4}\right )}{1536 c^{3} \left (e x +d \right )^{\frac {7}{2}} \left (-e x +d \right )^{2} e \,d^{5} \sqrt {c d}}\)	\(279\)

Input:

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

Output:

-1/1536*(c*(-e^2*x^2+d^2))^(1/2)/c^3*(-315*arctanh(1/2*(c*(-e*x+d))^(1/2)* 
2^(1/2)/(c*d)^(1/2))*2^(1/2)*e^4*x^4*(c*(-e*x+d))^(1/2)-630*arctanh(1/2*(c 
*(-e*x+d))^(1/2)*2^(1/2)/(c*d)^(1/2))*2^(1/2)*d*e^3*x^3*(c*(-e*x+d))^(1/2) 
+630*(c*d)^(1/2)*e^4*x^4+630*arctanh(1/2*(c*(-e*x+d))^(1/2)*2^(1/2)/(c*d)^ 
(1/2))*2^(1/2)*d^3*e*x*(c*(-e*x+d))^(1/2)+840*(c*d)^(1/2)*d*e^3*x^3+315*ar 
ctanh(1/2*(c*(-e*x+d))^(1/2)*2^(1/2)/(c*d)^(1/2))*2^(1/2)*d^4*(c*(-e*x+d)) 
^(1/2)-756*(c*d)^(1/2)*d^2*e^2*x^2-1224*(c*d)^(1/2)*d^3*e*x-2*(c*d)^(1/2)* 
d^4)/(e*x+d)^(7/2)/(-e*x+d)^2/e/d^5/(c*d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.94 \[ \int \frac {1}{(d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\left [\frac {315 \, \sqrt {2} {\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} + 2 \, d^{5} e x + d^{6}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, {\left (315 \, d e^{4} x^{4} + 420 \, d^{2} e^{3} x^{3} - 378 \, d^{3} e^{2} x^{2} - 612 \, d^{4} e x - d^{5}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{3072 \, {\left (c^{3} d^{6} e^{7} x^{6} + 2 \, c^{3} d^{7} e^{6} x^{5} - c^{3} d^{8} e^{5} x^{4} - 4 \, c^{3} d^{9} e^{4} x^{3} - c^{3} d^{10} e^{3} x^{2} + 2 \, c^{3} d^{11} e^{2} x + c^{3} d^{12} e\right )}}, \frac {315 \, \sqrt {2} {\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} + 2 \, d^{5} e x + d^{6}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{2 \, {\left (c d e x + c d^{2}\right )}}\right ) - 2 \, {\left (315 \, d e^{4} x^{4} + 420 \, d^{2} e^{3} x^{3} - 378 \, d^{3} e^{2} x^{2} - 612 \, d^{4} e x - d^{5}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{1536 \, {\left (c^{3} d^{6} e^{7} x^{6} + 2 \, c^{3} d^{7} e^{6} x^{5} - c^{3} d^{8} e^{5} x^{4} - 4 \, c^{3} d^{9} e^{4} x^{3} - c^{3} d^{10} e^{3} x^{2} + 2 \, c^{3} d^{11} e^{2} x + c^{3} d^{12} e\right )}}\right ] \] Input:

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

Output:

[1/3072*(315*sqrt(2)*(e^6*x^6 + 2*d*e^5*x^5 - d^2*e^4*x^4 - 4*d^3*e^3*x^3 
- d^4*e^2*x^2 + 2*d^5*e*x + d^6)*sqrt(c*d)*log(-(c*e^2*x^2 - 2*c*d*e*x - 3 
*c*d^2 + 2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(c*d)*sqrt(e*x + d))/(e^2* 
x^2 + 2*d*e*x + d^2)) - 4*(315*d*e^4*x^4 + 420*d^2*e^3*x^3 - 378*d^3*e^2*x 
^2 - 612*d^4*e*x - d^5)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d))/(c^3*d^6*e 
^7*x^6 + 2*c^3*d^7*e^6*x^5 - c^3*d^8*e^5*x^4 - 4*c^3*d^9*e^4*x^3 - c^3*d^1 
0*e^3*x^2 + 2*c^3*d^11*e^2*x + c^3*d^12*e), 1/1536*(315*sqrt(2)*(e^6*x^6 + 
 2*d*e^5*x^5 - d^2*e^4*x^4 - 4*d^3*e^3*x^3 - d^4*e^2*x^2 + 2*d^5*e*x + d^6 
)*sqrt(-c*d)*arctan(1/2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)*sqrt(e 
*x + d)/(c*d*e*x + c*d^2)) - 2*(315*d*e^4*x^4 + 420*d^2*e^3*x^3 - 378*d^3* 
e^2*x^2 - 612*d^4*e*x - d^5)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d))/(c^3* 
d^6*e^7*x^6 + 2*c^3*d^7*e^6*x^5 - c^3*d^8*e^5*x^4 - 4*c^3*d^9*e^4*x^3 - c^ 
3*d^10*e^3*x^2 + 2*c^3*d^11*e^2*x + c^3*d^12*e)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {315 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d} c^{2} d^{5}} + \frac {2 \, {\left (256 \, c^{4} d^{4} - 1152 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )} c^{3} d^{3} - 2772 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} c^{2} d^{2} - 1680 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} c d - 315 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{4}\right )}}{{\left (2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}\right )}^{3} c^{2} d^{5}}}{1536 \, e} \] Input:

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

Output:

1/1536*(315*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-(e*x + d)*c + 2*c*d)/sqrt(-c* 
d))/(sqrt(-c*d)*c^2*d^5) + 2*(256*c^4*d^4 - 1152*((e*x + d)*c - 2*c*d)*c^3 
*d^3 - 2772*((e*x + d)*c - 2*c*d)^2*c^2*d^2 - 1680*((e*x + d)*c - 2*c*d)^3 
*c*d - 315*((e*x + d)*c - 2*c*d)^4)/((2*sqrt(-(e*x + d)*c + 2*c*d)*c*d - ( 
-(e*x + d)*c + 2*c*d)^(3/2))^3*c^2*d^5))/e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (315 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) d^{4}+630 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) d^{3} e x -630 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) d \,e^{3} x^{3}-315 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) e^{4} x^{4}-315 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) d^{4}-630 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) d^{3} e x +630 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) d \,e^{3} x^{3}+315 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) e^{4} x^{4}+4 d^{5}+2448 d^{4} e x +1512 d^{3} e^{2} x^{2}-1680 d^{2} e^{3} x^{3}-1260 d \,e^{4} x^{4}\right )}{3072 \sqrt {-e x +d}\, c^{3} d^{6} e \left (-e^{4} x^{4}-2 d \,e^{3} x^{3}+2 d^{3} e x +d^{4}\right )} \] Input:

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

Output:

(sqrt(c)*(315*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) - sqrt(d)*sq 
rt(2))*d**4 + 630*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) - sqrt(d 
)*sqrt(2))*d**3*e*x - 630*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) 
- sqrt(d)*sqrt(2))*d*e**3*x**3 - 315*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqr 
t(d - e*x) - sqrt(d)*sqrt(2))*e**4*x**4 - 315*sqrt(d)*sqrt(d - e*x)*sqrt(2 
)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*d**4 - 630*sqrt(d)*sqrt(d - e*x)*sq 
rt(2)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*d**3*e*x + 630*sqrt(d)*sqrt(d - 
 e*x)*sqrt(2)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*d*e**3*x**3 + 315*sqrt( 
d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*e**4*x**4 + 
4*d**5 + 2448*d**4*e*x + 1512*d**3*e**2*x**2 - 1680*d**2*e**3*x**3 - 1260* 
d*e**4*x**4))/(3072*sqrt(d - e*x)*c**3*d**6*e*(d**4 + 2*d**3*e*x - 2*d*e** 
3*x**3 - e**4*x**4))