\(\int \frac {(c+d x)^{3/2} (c^2-d^2 x^2)^{3/2}}{(e x)^{19/2}} \, dx\) [939]

Optimal result

Integrand size = 33, antiderivative size = 306 \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=-\frac {2 c^2 \left (c^2-d^2 x^2\right )^{5/2}}{17 e (e x)^{17/2} (c+d x)^{5/2}}-\frac {42 c d \left (c^2-d^2 x^2\right )^{5/2}}{85 e^2 (e x)^{15/2} (c+d x)^{5/2}}-\frac {186 d^2 \left (c^2-d^2 x^2\right )^{5/2}}{221 e^3 (e x)^{13/2} (c+d x)^{5/2}}-\frac {1930 d^3 \left (c^2-d^2 x^2\right )^{5/2}}{2431 c e^4 (e x)^{11/2} (c+d x)^{5/2}}-\frac {3860 d^4 \left (c^2-d^2 x^2\right )^{5/2}}{7293 c^2 e^5 (e x)^{9/2} (c+d x)^{5/2}}-\frac {15440 d^5 \left (c^2-d^2 x^2\right )^{5/2}}{51051 c^3 e^6 (e x)^{7/2} (c+d x)^{5/2}}-\frac {6176 d^6 \left (c^2-d^2 x^2\right )^{5/2}}{51051 c^4 e^7 (e x)^{5/2} (c+d x)^{5/2}} \] Output:

-2/17*c^2*(-d^2*x^2+c^2)^(5/2)/e/(e*x)^(17/2)/(d*x+c)^(5/2)-42/85*c*d*(-d^ 
2*x^2+c^2)^(5/2)/e^2/(e*x)^(15/2)/(d*x+c)^(5/2)-186/221*d^2*(-d^2*x^2+c^2) 
^(5/2)/e^3/(e*x)^(13/2)/(d*x+c)^(5/2)-1930/2431*d^3*(-d^2*x^2+c^2)^(5/2)/c 
/e^4/(e*x)^(11/2)/(d*x+c)^(5/2)-3860/7293*d^4*(-d^2*x^2+c^2)^(5/2)/c^2/e^5 
/(e*x)^(9/2)/(d*x+c)^(5/2)-15440/51051*d^5*(-d^2*x^2+c^2)^(5/2)/c^3/e^6/(e 
*x)^(7/2)/(d*x+c)^(5/2)-6176/51051*d^6*(-d^2*x^2+c^2)^(5/2)/c^4/e^7/(e*x)^ 
(5/2)/(d*x+c)^(5/2)
 

Mathematica [A] (verified)

Time = 10.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.38 \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=-\frac {2 (c-d x)^2 \sqrt {c^2-d^2 x^2} \left (15015 c^6+63063 c^5 d x+107415 c^4 d^2 x^2+101325 c^3 d^3 x^3+67550 c^2 d^4 x^4+38600 c d^5 x^5+15440 d^6 x^6\right )}{255255 c^4 e^9 x^8 \sqrt {e x} \sqrt {c+d x}} \] Input:

Integrate[((c + d*x)^(3/2)*(c^2 - d^2*x^2)^(3/2))/(e*x)^(19/2),x]
 

Output:

(-2*(c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(15015*c^6 + 63063*c^5*d*x + 107415*c^ 
4*d^2*x^2 + 101325*c^3*d^3*x^3 + 67550*c^2*d^4*x^4 + 38600*c*d^5*x^5 + 154 
40*d^6*x^6))/(255255*c^4*e^9*x^8*Sqrt[e*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {586, 108, 27, 167, 27, 167, 27, 162, 55, 55, 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.

\(\displaystyle \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx\)

\(\Big \downarrow \) 586

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \int \frac {(c-d x)^{3/2} (c+d x)^3}{(e x)^{19/2}}dx}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {2 \int \frac {3 d (c-3 d x) \sqrt {c-d x} (c+d x)^2}{2 (e x)^{17/2}}dx}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \int \frac {(c-3 d x) \sqrt {c-d x} (c+d x)^2}{(e x)^{17/2}}dx}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (\frac {2 \int -\frac {c d e (55 c-53 d x) (c+d x)^2}{2 (e x)^{15/2} \sqrt {c-d x}}dx}{15 c e^2}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \int \frac {(55 c-53 d x) (c+d x)^2}{(e x)^{15/2} \sqrt {c-d x}}dx}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {2 \int \frac {c d e (191 c-249 d x) (c+d x)}{2 (e x)^{13/2} \sqrt {c-d x}}dx}{13 c e^2}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {d \int \frac {(191 c-249 d x) (c+d x)}{(e x)^{13/2} \sqrt {c-d x}}dx}{13 e}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {d \left (-\frac {4825 d^2 \int \frac {1}{(e x)^{9/2} \sqrt {c-d x}}dx}{33 e^2}-\frac {2 \sqrt {c-d x} (573 c+424 d x)}{33 e (e x)^{11/2}}\right )}{13 e}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {d \left (-\frac {4825 d^2 \left (\frac {6 d \int \frac {1}{(e x)^{7/2} \sqrt {c-d x}}dx}{7 c e}-\frac {2 \sqrt {c-d x}}{7 c e (e x)^{7/2}}\right )}{33 e^2}-\frac {2 \sqrt {c-d x} (573 c+424 d x)}{33 e (e x)^{11/2}}\right )}{13 e}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {d \left (-\frac {4825 d^2 \left (\frac {6 d \left (\frac {4 d \int \frac {1}{(e x)^{5/2} \sqrt {c-d x}}dx}{5 c e}-\frac {2 \sqrt {c-d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c-d x}}{7 c e (e x)^{7/2}}\right )}{33 e^2}-\frac {2 \sqrt {c-d x} (573 c+424 d x)}{33 e (e x)^{11/2}}\right )}{13 e}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {d \left (-\frac {4825 d^2 \left (\frac {6 d \left (\frac {4 d \left (\frac {2 d \int \frac {1}{(e x)^{3/2} \sqrt {c-d x}}dx}{3 c e}-\frac {2 \sqrt {c-d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c-d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c-d x}}{7 c e (e x)^{7/2}}\right )}{33 e^2}-\frac {2 \sqrt {c-d x} (573 c+424 d x)}{33 e (e x)^{11/2}}\right )}{13 e}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} \left (\frac {3 d \left (-\frac {d \left (\frac {d \left (-\frac {4825 d^2 \left (\frac {6 d \left (\frac {4 d \left (-\frac {4 d \sqrt {c-d x}}{3 c^2 e^2 \sqrt {e x}}-\frac {2 \sqrt {c-d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c-d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c-d x}}{7 c e (e x)^{7/2}}\right )}{33 e^2}-\frac {2 \sqrt {c-d x} (573 c+424 d x)}{33 e (e x)^{11/2}}\right )}{13 e}-\frac {110 \sqrt {c-d x} (c+d x)^2}{13 e (e x)^{13/2}}\right )}{15 e}-\frac {2 \sqrt {c-d x} (c+d x)^3}{15 e (e x)^{15/2}}\right )}{17 e}-\frac {2 (c-d x)^{3/2} (c+d x)^3}{17 e (e x)^{17/2}}\right )}{\sqrt {c-d x} \sqrt {c+d x}}\)

Input:

Int[((c + d*x)^(3/2)*(c^2 - d^2*x^2)^(3/2))/(e*x)^(19/2),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*((-2*(c - d*x)^(3/2)*(c + d*x)^3)/(17*e*(e*x)^(17/2)) 
 + (3*d*((-2*Sqrt[c - d*x]*(c + d*x)^3)/(15*e*(e*x)^(15/2)) - (d*((-110*Sq 
rt[c - d*x]*(c + d*x)^2)/(13*e*(e*x)^(13/2)) + (d*((-2*Sqrt[c - d*x]*(573* 
c + 424*d*x))/(33*e*(e*x)^(11/2)) - (4825*d^2*((-2*Sqrt[c - d*x])/(7*c*e*( 
e*x)^(7/2)) + (6*d*((-2*Sqrt[c - d*x])/(5*c*e*(e*x)^(5/2)) + (4*d*((-2*Sqr 
t[c - d*x])/(3*c*e*(e*x)^(3/2)) - (4*d*Sqrt[c - d*x])/(3*c^2*e^2*Sqrt[e*x] 
)))/(5*c*e)))/(7*c*e)))/(33*e^2)))/(13*e)))/(15*e)))/(17*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 108

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

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

rule 167

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*((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 - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

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 = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.33

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) x \left (15440 d^{6} x^{6}+38600 c \,d^{5} x^{5}+67550 c^{2} d^{4} x^{4}+101325 d^{3} c^{3} x^{3}+107415 c^{4} d^{2} x^{2}+63063 c^{5} d x +15015 c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{255255 c^{4} \left (d x +c \right )^{\frac {3}{2}} \left (e x \right )^{\frac {19}{2}}}\)	\(102\)
orering	\(-\frac {2 \left (-d x +c \right ) x \left (15440 d^{6} x^{6}+38600 c \,d^{5} x^{5}+67550 c^{2} d^{4} x^{4}+101325 d^{3} c^{3} x^{3}+107415 c^{4} d^{2} x^{2}+63063 c^{5} d x +15015 c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{255255 c^{4} \left (d x +c \right )^{\frac {3}{2}} \left (e x \right )^{\frac {19}{2}}}\)	\(102\)
default	\(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right ) \left (-15440 d^{7} x^{7}-23160 c \,d^{6} x^{6}-28950 c^{2} d^{5} x^{5}-33775 d^{4} c^{3} x^{4}-6090 d^{3} c^{4} x^{3}+44352 c^{5} d^{2} x^{2}+48048 c^{6} d x +15015 c^{7}\right )}{255255 \sqrt {d x +c}\, e^{9} x^{8} \sqrt {e x}\, c^{4}}\)	\(118\)

Input:

int((d*x+c)^(3/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(19/2),x,method=_RETURNVERBOS 
E)
 

Output:

-2/255255*(-d*x+c)*x*(15440*d^6*x^6+38600*c*d^5*x^5+67550*c^2*d^4*x^4+1013 
25*c^3*d^3*x^3+107415*c^4*d^2*x^2+63063*c^5*d*x+15015*c^6)*(-d^2*x^2+c^2)^ 
(3/2)/c^4/(d*x+c)^(3/2)/(e*x)^(19/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.45 \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=-\frac {2 \, {\left (15440 \, d^{8} x^{8} + 7720 \, c d^{7} x^{7} + 5790 \, c^{2} d^{6} x^{6} + 4825 \, c^{3} d^{5} x^{5} - 27685 \, c^{4} d^{4} x^{4} - 50442 \, c^{5} d^{3} x^{3} - 3696 \, c^{6} d^{2} x^{2} + 33033 \, c^{7} d x + 15015 \, c^{8}\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {e x}}{255255 \, {\left (c^{4} d e^{10} x^{10} + c^{5} e^{10} x^{9}\right )}} \] Input:

integrate((d*x+c)^(3/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(19/2),x, algorithm="fr 
icas")
 

Output:

-2/255255*(15440*d^8*x^8 + 7720*c*d^7*x^7 + 5790*c^2*d^6*x^6 + 4825*c^3*d^ 
5*x^5 - 27685*c^4*d^4*x^4 - 50442*c^5*d^3*x^3 - 3696*c^6*d^2*x^2 + 33033*c 
^7*d*x + 15015*c^8)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x)/(c^4*d*e^ 
10*x^10 + c^5*e^10*x^9)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(3/2)*(-d**2*x**2+c**2)**(3/2)/(e*x)**(19/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {19}{2}}} \,d x } \] Input:

integrate((d*x+c)^(3/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(19/2),x, algorithm="ma 
xima")
 

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^(3/2)/(e*x)^(19/2), x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.57 \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=-\frac {2 \, {\left (408408 \, c^{2} d^{17} e^{8} + {\left (1137708 \, c d^{17} e^{8} + 5 \, {\left (286858 \, d^{17} e^{8} + 193 \, {\left (\frac {1105 \, d^{17} e^{8}}{c} + 2 \, {\left (\frac {255 \, d^{17} e^{8}}{c^{2}} + 4 \, {\left (\frac {2 \, {\left (d x - c\right )} d^{17} e^{8}}{c^{4}} + \frac {17 \, d^{17} e^{8}}{c^{3}}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}\right )} {\left (d x - c\right )}^{2} \sqrt {-d x + c} d}{255255 \, {\left ({\left (d x - c\right )} d e + c d e\right )}^{\frac {17}{2}} e^{9} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(19/2),x, algorithm="gi 
ac")
 

Output:

-2/255255*(408408*c^2*d^17*e^8 + (1137708*c*d^17*e^8 + 5*(286858*d^17*e^8 
+ 193*(1105*d^17*e^8/c + 2*(255*d^17*e^8/c^2 + 4*(2*(d*x - c)*d^17*e^8/c^4 
 + 17*d^17*e^8/c^3)*(d*x - c))*(d*x - c))*(d*x - c))*(d*x - c))*(d*x - c)) 
*(d*x - c)^2*sqrt(-d*x + c)*d/(((d*x - c)*d*e + c*d*e)^(17/2)*e^9*abs(d))
 

Mupad [B] (verification not implemented)

Time = 7.48 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.71 \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=-\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {2\,c^4\,\sqrt {c+d\,x}}{17\,d\,e^9}-\frac {1582\,d^3\,x^4\,\sqrt {c+d\,x}}{7293\,e^9}+\frac {22\,c^3\,x\,\sqrt {c+d\,x}}{85\,e^9}-\frac {32\,c^2\,d\,x^2\,\sqrt {c+d\,x}}{1105\,e^9}-\frac {4804\,c\,d^2\,x^3\,\sqrt {c+d\,x}}{12155\,e^9}+\frac {1930\,d^4\,x^5\,\sqrt {c+d\,x}}{51051\,c\,e^9}+\frac {772\,d^5\,x^6\,\sqrt {c+d\,x}}{17017\,c^2\,e^9}+\frac {3088\,d^6\,x^7\,\sqrt {c+d\,x}}{51051\,c^3\,e^9}+\frac {6176\,d^7\,x^8\,\sqrt {c+d\,x}}{51051\,c^4\,e^9}\right )}{x^9\,\sqrt {e\,x}+\frac {c\,x^8\,\sqrt {e\,x}}{d}} \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(c + d*x)^(3/2))/(e*x)^(19/2),x)
 

Output:

-((c^2 - d^2*x^2)^(1/2)*((2*c^4*(c + d*x)^(1/2))/(17*d*e^9) - (1582*d^3*x^ 
4*(c + d*x)^(1/2))/(7293*e^9) + (22*c^3*x*(c + d*x)^(1/2))/(85*e^9) - (32* 
c^2*d*x^2*(c + d*x)^(1/2))/(1105*e^9) - (4804*c*d^2*x^3*(c + d*x)^(1/2))/( 
12155*e^9) + (1930*d^4*x^5*(c + d*x)^(1/2))/(51051*c*e^9) + (772*d^5*x^6*( 
c + d*x)^(1/2))/(17017*c^2*e^9) + (3088*d^6*x^7*(c + d*x)^(1/2))/(51051*c^ 
3*e^9) + (6176*d^7*x^8*(c + d*x)^(1/2))/(51051*c^4*e^9)))/(x^9*(e*x)^(1/2) 
 + (c*x^8*(e*x)^(1/2))/d)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.62 \[ \int \frac {(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{(e x)^{19/2}} \, dx=\frac {2 \sqrt {e}\, \left (-15015 \sqrt {x}\, \sqrt {-d x +c}\, c^{8}-33033 \sqrt {x}\, \sqrt {-d x +c}\, c^{7} d x +3696 \sqrt {x}\, \sqrt {-d x +c}\, c^{6} d^{2} x^{2}+50442 \sqrt {x}\, \sqrt {-d x +c}\, c^{5} d^{3} x^{3}+27685 \sqrt {x}\, \sqrt {-d x +c}\, c^{4} d^{4} x^{4}-4825 \sqrt {x}\, \sqrt {-d x +c}\, c^{3} d^{5} x^{5}-5790 \sqrt {x}\, \sqrt {-d x +c}\, c^{2} d^{6} x^{6}-7720 \sqrt {x}\, \sqrt {-d x +c}\, c \,d^{7} x^{7}-15440 \sqrt {x}\, \sqrt {-d x +c}\, d^{8} x^{8}+15440 \sqrt {d}\, d^{8} i \,x^{9}\right )}{255255 c^{4} e^{10} x^{9}} \] Input:

int((d*x+c)^(3/2)*(-d^2*x^2+c^2)^(3/2)/(e*x)^(19/2),x)
 

Output:

(2*sqrt(e)*( - 15015*sqrt(x)*sqrt(c - d*x)*c**8 - 33033*sqrt(x)*sqrt(c - d 
*x)*c**7*d*x + 3696*sqrt(x)*sqrt(c - d*x)*c**6*d**2*x**2 + 50442*sqrt(x)*s 
qrt(c - d*x)*c**5*d**3*x**3 + 27685*sqrt(x)*sqrt(c - d*x)*c**4*d**4*x**4 - 
 4825*sqrt(x)*sqrt(c - d*x)*c**3*d**5*x**5 - 5790*sqrt(x)*sqrt(c - d*x)*c* 
*2*d**6*x**6 - 7720*sqrt(x)*sqrt(c - d*x)*c*d**7*x**7 - 15440*sqrt(x)*sqrt 
(c - d*x)*d**8*x**8 + 15440*sqrt(d)*d**8*i*x**9))/(255255*c**4*e**10*x**9)