\(\int \frac {(c d^2-c e^2 x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [178]

Optimal result

Integrand size = 29, antiderivative size = 133 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {3 c \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac {3 \sqrt {2} c^{3/2} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {c \sqrt {c \left (d^2-e^2 x^2\right )} \left (-\frac {2 (2 d+e x)}{(d+e x)^{3/2}}+\frac {3 \sqrt {2} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\right )}{e} \] Input:

Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {465, 466, 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.

\(\displaystyle \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 465

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

\(\Big \downarrow \) 466

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

Input:

Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]
 

Output:

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

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 465

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

rule 466

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] - Simp[2*b*c*(p/(d^ 
2*(n + 2*p + 1)))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; Fr 
eeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LeQ[-2, n, 0 
] || EqQ[n + p + 1, 0]) && NeQ[n + 2*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 = 142, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {\sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, c \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c d e x +3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c \,d^{2}-2 e x \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}-4 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\, d \right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\)	\(142\)
risch	\(-\frac {2 \left (-e x +d \right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2}}{e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}}-\frac {4 d \left (-\frac {\sqrt {-c e x +c d}}{2 \left (-c e x -c d \right )}-\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c e x +c d}\, \sqrt {2}}{2 \sqrt {c d}}\right )}{4 \sqrt {c d}}\right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2}}{e \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}}\)	\(195\)

Input:

int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.31 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (c e x + 2 \, c d\right )} \sqrt {e x + d}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}}, -\frac {3 \, \sqrt {2} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{2 \, {\left (c d e x + c d^{2}\right )}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (c e x + 2 \, c d\right )} \sqrt {e x + d}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right ] \] Input:

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(7/2),x)
 

Output:

Integral((-c*(-d + e*x)*(d + e*x))**(3/2)/(d + e*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(7/2), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} c^{2} d \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + 2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c + \frac {2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d}{e x + d}}{e} \] Input:

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int((c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x)
 

Output:

int((c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {c}\, c \left (-16 \sqrt {-e x +d}\, d -8 \sqrt {-e x +d}\, e x -12 \sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e x +d}}{\sqrt {d}\, \sqrt {2}}\right )}{2}\right )\right ) d -12 \sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e x +d}}{\sqrt {d}\, \sqrt {2}}\right )}{2}\right )\right ) e x +9 \sqrt {d}\, \sqrt {2}\, d +9 \sqrt {d}\, \sqrt {2}\, e x \right )}{4 e \left (e x +d \right )} \] Input:

int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(7/2),x)
 

Output:

(sqrt(c)*c*( - 16*sqrt(d - e*x)*d - 8*sqrt(d - e*x)*e*x - 12*sqrt(d)*sqrt( 
2)*log(tan(asin(sqrt(d + e*x)/(sqrt(d)*sqrt(2)))/2))*d - 12*sqrt(d)*sqrt(2 
)*log(tan(asin(sqrt(d + e*x)/(sqrt(d)*sqrt(2)))/2))*e*x + 9*sqrt(d)*sqrt(2 
)*d + 9*sqrt(d)*sqrt(2)*e*x))/(4*e*(d + e*x))