\(\int \frac {\sqrt {c+d x} (c^2-d^2 x^2)^{3/2}}{(e x)^{13/2}} \, dx\) [925]

Optimal result

Integrand size = 33, antiderivative size = 171 \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=-\frac {2 c \left (c^2-d^2 x^2\right )^{5/2}}{11 e (e x)^{11/2} (c+d x)^{5/2}}-\frac {56 d \left (c^2-d^2 x^2\right )^{5/2}}{99 e^2 (e x)^{9/2} (c+d x)^{5/2}}-\frac {422 d^2 \left (c^2-d^2 x^2\right )^{5/2}}{693 c e^3 (e x)^{7/2} (c+d x)^{5/2}}-\frac {844 d^3 \left (c^2-d^2 x^2\right )^{5/2}}{3465 c^2 e^4 (e x)^{5/2} (c+d x)^{5/2}} \] Output:

-2/11*c*(-d^2*x^2+c^2)^(5/2)/e/(e*x)^(11/2)/(d*x+c)^(5/2)-56/99*d*(-d^2*x^ 
2+c^2)^(5/2)/e^2/(e*x)^(9/2)/(d*x+c)^(5/2)-422/693*d^2*(-d^2*x^2+c^2)^(5/2 
)/c/e^3/(e*x)^(7/2)/(d*x+c)^(5/2)-844/3465*d^3*(-d^2*x^2+c^2)^(5/2)/c^2/e^ 
4/(e*x)^(5/2)/(d*x+c)^(5/2)
 

Mathematica [A] (verified)

Time = 6.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=-\frac {2 \sqrt {e x} (c-d x)^2 \sqrt {c^2-d^2 x^2} \left (315 c^3+980 c^2 d x+1055 c d^2 x^2+422 d^3 x^3\right )}{3465 c^2 e^7 x^6 \sqrt {c+d x}} \] Input:

Integrate[(Sqrt[c + d*x]*(c^2 - d^2*x^2)^(3/2))/(e*x)^(13/2),x]
 

Output:

(-2*Sqrt[e*x]*(c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(315*c^3 + 980*c^2*d*x + 105 
5*c*d^2*x^2 + 422*d^3*x^3))/(3465*c^2*e^7*x^6*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {586, 100, 27, 87, 55, 48}

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.

\(\displaystyle \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx\)

\(\Big \downarrow \) 586

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \int \frac {(c-d x)^{3/2} (c+d x)^2}{(e x)^{13/2}}dx}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {2 \int \frac {c d e^2 (c-d x)^{3/2} (28 c+11 d x)}{2 (e x)^{11/2}}dx}{11 c e^3}-\frac {2 c (c-d x)^{5/2}}{11 e (e x)^{11/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {d \int \frac {(c-d x)^{3/2} (28 c+11 d x)}{(e x)^{11/2}}dx}{11 e}-\frac {2 c (c-d x)^{5/2}}{11 e (e x)^{11/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {d \left (\frac {211 d \int \frac {(c-d x)^{3/2}}{(e x)^{9/2}}dx}{9 e}-\frac {56 (c-d x)^{5/2}}{9 e (e x)^{9/2}}\right )}{11 e}-\frac {2 c (c-d x)^{5/2}}{11 e (e x)^{11/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {d \left (\frac {211 d \left (\frac {2 d \int \frac {(c-d x)^{3/2}}{(e x)^{7/2}}dx}{7 c e}-\frac {2 (c-d x)^{5/2}}{7 c e (e x)^{7/2}}\right )}{9 e}-\frac {56 (c-d x)^{5/2}}{9 e (e x)^{9/2}}\right )}{11 e}-\frac {2 c (c-d x)^{5/2}}{11 e (e x)^{11/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {d \left (\frac {211 d \left (-\frac {4 d (c-d x)^{5/2}}{35 c^2 e^2 (e x)^{5/2}}-\frac {2 (c-d x)^{5/2}}{7 c e (e x)^{7/2}}\right )}{9 e}-\frac {56 (c-d x)^{5/2}}{9 e (e x)^{9/2}}\right )}{11 e}-\frac {2 c (c-d x)^{5/2}}{11 e (e x)^{11/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

Input:

Int[(Sqrt[c + d*x]*(c^2 - d^2*x^2)^(3/2))/(e*x)^(13/2),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*((-2*c*(c - d*x)^(5/2))/(11*e*(e*x)^(11/2)) + (d*((-5 
6*(c - d*x)^(5/2))/(9*e*(e*x)^(9/2)) + (211*d*((-2*(c - d*x)^(5/2))/(7*c*e 
*(e*x)^(7/2)) - (4*d*(c - d*x)^(5/2))/(35*c^2*e^2*(e*x)^(5/2))))/(9*e)))/( 
11*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 48

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

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

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

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 = 69, normalized size of antiderivative = 0.40

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) x \left (422 d^{3} x^{3}+1055 c \,d^{2} x^{2}+980 c^{2} d x +315 c^{3}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3465 c^{2} \left (d x +c \right )^{\frac {3}{2}} \left (e x \right )^{\frac {13}{2}}}\)	\(69\)
orering	\(-\frac {2 \left (-d x +c \right ) x \left (422 d^{3} x^{3}+1055 c \,d^{2} x^{2}+980 c^{2} d x +315 c^{3}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3465 c^{2} \left (d x +c \right )^{\frac {3}{2}} \left (e x \right )^{\frac {13}{2}}}\)	\(69\)
default	\(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right ) \left (-422 d^{4} x^{4}-633 c \,d^{3} x^{3}+75 d^{2} c^{2} x^{2}+665 c^{3} d x +315 c^{4}\right )}{3465 \sqrt {d x +c}\, e^{6} x^{5} \sqrt {e x}\, c^{2}}\)	\(85\)

Input:

int((d*x+c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(13/2),x,method=_RETURNVERBOS 
E)
 

Output:

-2/3465*(-d*x+c)*x*(422*d^3*x^3+1055*c*d^2*x^2+980*c^2*d*x+315*c^3)*(-d^2* 
x^2+c^2)^(3/2)/c^2/(d*x+c)^(3/2)/(e*x)^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=-\frac {2 \, {\left (422 \, d^{5} x^{5} + 211 \, c d^{4} x^{4} - 708 \, c^{2} d^{3} x^{3} - 590 \, c^{3} d^{2} x^{2} + 350 \, c^{4} d x + 315 \, c^{5}\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {e x}}{3465 \, {\left (c^{2} d e^{7} x^{7} + c^{3} e^{7} x^{6}\right )}} \] Input:

integrate((d*x+c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(13/2),x, algorithm="fr 
icas")
 

Output:

-2/3465*(422*d^5*x^5 + 211*c*d^4*x^4 - 708*c^2*d^3*x^3 - 590*c^3*d^2*x^2 + 
 350*c^4*d*x + 315*c^5)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x)/(c^2* 
d*e^7*x^7 + c^3*e^7*x^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(-d**2*x**2+c**2)**(3/2)/(e*x)**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} \sqrt {d x + c}}{\left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(13/2),x, algorithm="ma 
xima")
 

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*sqrt(d*x + c)/(e*x)^(13/2), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=-\frac {2 \, {\left (2772 \, c d^{11} e^{5} + {\left (4356 \, d^{11} e^{5} + 211 \, {\left (\frac {2 \, {\left (d x - c\right )} d^{11} e^{5}}{c^{2}} + \frac {11 \, d^{11} e^{5}}{c}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}^{2} \sqrt {-d x + c} d}{3465 \, {\left ({\left (d x - c\right )} d e + c d e\right )}^{\frac {11}{2}} e^{6} {\left | d \right |}} \] Input:

integrate((d*x+c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(13/2),x, algorithm="gi 
ac")
 

Output:

-2/3465*(2772*c*d^11*e^5 + (4356*d^11*e^5 + 211*(2*(d*x - c)*d^11*e^5/c^2 
+ 11*d^11*e^5/c)*(d*x - c))*(d*x - c))*(d*x - c)^2*sqrt(-d*x + c)*d/(((d*x 
 - c)*d*e + c*d*e)^(11/2)*e^6*abs(d))
 

Mupad [B] (verification not implemented)

Time = 7.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=-\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {2\,c^3\,\sqrt {c+d\,x}}{11\,d\,e^6}-\frac {472\,d^2\,x^3\,\sqrt {c+d\,x}}{1155\,e^6}+\frac {20\,c^2\,x\,\sqrt {c+d\,x}}{99\,e^6}-\frac {236\,c\,d\,x^2\,\sqrt {c+d\,x}}{693\,e^6}+\frac {422\,d^3\,x^4\,\sqrt {c+d\,x}}{3465\,c\,e^6}+\frac {844\,d^4\,x^5\,\sqrt {c+d\,x}}{3465\,c^2\,e^6}\right )}{x^6\,\sqrt {e\,x}+\frac {c\,x^5\,\sqrt {e\,x}}{d}} \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(c + d*x)^(1/2))/(e*x)^(13/2),x)
 

Output:

-((c^2 - d^2*x^2)^(1/2)*((2*c^3*(c + d*x)^(1/2))/(11*d*e^6) - (472*d^2*x^3 
*(c + d*x)^(1/2))/(1155*e^6) + (20*c^2*x*(c + d*x)^(1/2))/(99*e^6) - (236* 
c*d*x^2*(c + d*x)^(1/2))/(693*e^6) + (422*d^3*x^4*(c + d*x)^(1/2))/(3465*c 
*e^6) + (844*d^4*x^5*(c + d*x)^(1/2))/(3465*c^2*e^6)))/(x^6*(e*x)^(1/2) + 
(c*x^5*(e*x)^(1/2))/d)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{13/2}} \, dx=\frac {2 \sqrt {e}\, \left (-315 \sqrt {x}\, \sqrt {-d x +c}\, c^{5}-350 \sqrt {x}\, \sqrt {-d x +c}\, c^{4} d x +590 \sqrt {x}\, \sqrt {-d x +c}\, c^{3} d^{2} x^{2}+708 \sqrt {x}\, \sqrt {-d x +c}\, c^{2} d^{3} x^{3}-211 \sqrt {x}\, \sqrt {-d x +c}\, c \,d^{4} x^{4}-422 \sqrt {x}\, \sqrt {-d x +c}\, d^{5} x^{5}+422 \sqrt {d}\, d^{5} i \,x^{6}\right )}{3465 c^{2} e^{7} x^{6}} \] Input:

int((d*x+c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(13/2),x)
 

Output:

(2*sqrt(e)*( - 315*sqrt(x)*sqrt(c - d*x)*c**5 - 350*sqrt(x)*sqrt(c - d*x)* 
c**4*d*x + 590*sqrt(x)*sqrt(c - d*x)*c**3*d**2*x**2 + 708*sqrt(x)*sqrt(c - 
 d*x)*c**2*d**3*x**3 - 211*sqrt(x)*sqrt(c - d*x)*c*d**4*x**4 - 422*sqrt(x) 
*sqrt(c - d*x)*d**5*x**5 + 422*sqrt(d)*d**5*i*x**6))/(3465*c**2*e**7*x**6)