\(\int (d+e x)^{5/2} (c d^2-c e^2 x^2)^{3/2} \, dx\) [172]

Optimal result

Integrand size = 29, antiderivative size = 201 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e} \] Output:

-4096/15015*d^4*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(5/2)-1024/3003*d^3*( 
-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(3/2)-128/429*d^2*(-c*e^2*x^2+c*d^2)^( 
5/2)/c/e/(e*x+d)^(1/2)-32/143*d*(e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(5/2)/c/e 
-2/13*(e*x+d)^(3/2)*(-c*e^2*x^2+c*d^2)^(5/2)/c/e
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.42 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 c (d-e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (9683 d^4+16700 d^3 e x+14210 d^2 e^2 x^2+6300 d e^3 x^3+1155 e^4 x^4\right )}{15015 e \sqrt {d+e x}} \] Input:

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

Output:

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(9683*d^4 + 16700*d^3*e*x + 1421 
0*d^2*e^2*x^2 + 6300*d*e^3*x^3 + 1155*e^4*x^4))/(15015*e*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {459, 459, 459, 459, 458}

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.

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

\(\Big \downarrow \) 459

\(\displaystyle \frac {16}{13} d \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}dx-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {16}{13} d \left (\frac {12}{11} d \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}dx-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}\right )-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {16}{13} d \left (\frac {12}{11} d \left (\frac {8}{9} d \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}}dx-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}\right )-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {16}{13} d \left (\frac {12}{11} d \left (\frac {8}{9} d \left (\frac {4}{7} d \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}}dx-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}\right )-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\)

\(\Big \downarrow \) 458

\(\displaystyle \frac {16}{13} d \left (\frac {12}{11} d \left (\frac {8}{9} d \left (-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}}\right )-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}\right )-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\)

Input:

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

Output:

(-2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(5/2))/(13*c*e) + (16*d*((-2*Sqrt[ 
d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2))/(11*c*e) + (12*d*((-2*(c*d^2 - c*e^2*x 
^2)^(5/2))/(9*c*e*Sqrt[d + e*x]) + (8*d*((-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/ 
(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(7*c*e*(d + e*x)^ 
(3/2))))/9))/11))/13
 

Defintions of rubi rules used

rule 458

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

rule 459

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.38

method	result	size
gosper	\(-\frac {2 \left (-e x +d \right ) \left (1155 e^{4} x^{4}+6300 d \,e^{3} x^{3}+14210 d^{2} e^{2} x^{2}+16700 d^{3} e x +9683 d^{4}\right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{15015 e \left (e x +d \right )^{\frac {3}{2}}}\)	\(77\)
orering	\(-\frac {2 \left (-e x +d \right ) \left (1155 e^{4} x^{4}+6300 d \,e^{3} x^{3}+14210 d^{2} e^{2} x^{2}+16700 d^{3} e x +9683 d^{4}\right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{15015 e \left (e x +d \right )^{\frac {3}{2}}}\)	\(77\)
default	\(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, c \left (-e x +d \right )^{2} \left (1155 e^{4} x^{4}+6300 d \,e^{3} x^{3}+14210 d^{2} e^{2} x^{2}+16700 d^{3} e x +9683 d^{4}\right )}{15015 \sqrt {e x +d}\, e}\)	\(79\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2} \left (1155 e^{6} x^{6}+3990 d \,e^{5} x^{5}+2765 d^{2} e^{4} x^{4}-5420 d^{3} e^{3} x^{3}-9507 d^{4} e^{2} x^{2}-2666 d^{5} e x +9683 d^{6}\right ) \left (-e x +d \right )}{15015 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(140\)

Input:

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

Output:

-2/15015*(-e*x+d)*(1155*e^4*x^4+6300*d*e^3*x^3+14210*d^2*e^2*x^2+16700*d^3 
*e*x+9683*d^4)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.53 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (1155 \, c e^{6} x^{6} + 3990 \, c d e^{5} x^{5} + 2765 \, c d^{2} e^{4} x^{4} - 5420 \, c d^{3} e^{3} x^{3} - 9507 \, c d^{4} e^{2} x^{2} - 2666 \, c d^{5} e x + 9683 \, c d^{6}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{15015 \, {\left (e^{2} x + d e\right )}} \] Input:

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

Output:

-2/15015*(1155*c*e^6*x^6 + 3990*c*d*e^5*x^5 + 2765*c*d^2*e^4*x^4 - 5420*c* 
d^3*e^3*x^3 - 9507*c*d^4*e^2*x^2 - 2666*c*d^5*e*x + 9683*c*d^6)*sqrt(-c*e^ 
2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.55 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (1155 \, c^{\frac {3}{2}} e^{6} x^{6} + 3990 \, c^{\frac {3}{2}} d e^{5} x^{5} + 2765 \, c^{\frac {3}{2}} d^{2} e^{4} x^{4} - 5420 \, c^{\frac {3}{2}} d^{3} e^{3} x^{3} - 9507 \, c^{\frac {3}{2}} d^{4} e^{2} x^{2} - 2666 \, c^{\frac {3}{2}} d^{5} e x + 9683 \, c^{\frac {3}{2}} d^{6}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{15015 \, {\left (e^{2} x + d e\right )}} \] Input:

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

Output:

-2/15015*(1155*c^(3/2)*e^6*x^6 + 3990*c^(3/2)*d*e^5*x^5 + 2765*c^(3/2)*d^2 
*e^4*x^4 - 5420*c^(3/2)*d^3*e^3*x^3 - 9507*c^(3/2)*d^4*e^2*x^2 - 2666*c^(3 
/2)*d^5*e*x + 9683*c^(3/2)*d^6)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (171) = 342\).

Time = 0.14 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.42 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (45045 \, \sqrt {-c e x + c d} c d^{6} + 30030 \, {\left (3 \, \sqrt {-c e x + c d} c d - {\left (-c e x + c d\right )}^{\frac {3}{2}}\right )} d^{5} - \frac {3003 \, {\left (15 \, \sqrt {-c e x + c d} c^{2} d^{2} - 10 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c d + 3 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d}\right )} d^{4}}{c} - \frac {5148 \, {\left (35 \, \sqrt {-c e x + c d} c^{3} d^{3} - 35 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c^{2} d^{2} + 21 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d} c d + 5 \, {\left (c e x - c d\right )}^{3} \sqrt {-c e x + c d}\right )} d^{3}}{c^{2}} - \frac {143 \, {\left (315 \, \sqrt {-c e x + c d} c^{4} d^{4} - 420 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c^{3} d^{3} + 378 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d} c^{2} d^{2} + 180 \, {\left (c e x - c d\right )}^{3} \sqrt {-c e x + c d} c d + 35 \, {\left (c e x - c d\right )}^{4} \sqrt {-c e x + c d}\right )} d^{2}}{c^{3}} + \frac {130 \, {\left (693 \, \sqrt {-c e x + c d} c^{5} d^{5} - 1155 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c^{4} d^{4} + 1386 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d} c^{3} d^{3} + 990 \, {\left (c e x - c d\right )}^{3} \sqrt {-c e x + c d} c^{2} d^{2} + 385 \, {\left (c e x - c d\right )}^{4} \sqrt {-c e x + c d} c d + 63 \, {\left (c e x - c d\right )}^{5} \sqrt {-c e x + c d}\right )} d}{c^{4}} + \frac {15 \, {\left (3003 \, \sqrt {-c e x + c d} c^{6} d^{6} - 6006 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c^{5} d^{5} + 9009 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d} c^{4} d^{4} + 8580 \, {\left (c e x - c d\right )}^{3} \sqrt {-c e x + c d} c^{3} d^{3} + 5005 \, {\left (c e x - c d\right )}^{4} \sqrt {-c e x + c d} c^{2} d^{2} + 1638 \, {\left (c e x - c d\right )}^{5} \sqrt {-c e x + c d} c d + 231 \, {\left (c e x - c d\right )}^{6} \sqrt {-c e x + c d}\right )}}{c^{5}}\right )}}{45045 \, e} \] Input:

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

Output:

-2/45045*(45045*sqrt(-c*e*x + c*d)*c*d^6 + 30030*(3*sqrt(-c*e*x + c*d)*c*d 
 - (-c*e*x + c*d)^(3/2))*d^5 - 3003*(15*sqrt(-c*e*x + c*d)*c^2*d^2 - 10*(- 
c*e*x + c*d)^(3/2)*c*d + 3*(c*e*x - c*d)^2*sqrt(-c*e*x + c*d))*d^4/c - 514 
8*(35*sqrt(-c*e*x + c*d)*c^3*d^3 - 35*(-c*e*x + c*d)^(3/2)*c^2*d^2 + 21*(c 
*e*x - c*d)^2*sqrt(-c*e*x + c*d)*c*d + 5*(c*e*x - c*d)^3*sqrt(-c*e*x + c*d 
))*d^3/c^2 - 143*(315*sqrt(-c*e*x + c*d)*c^4*d^4 - 420*(-c*e*x + c*d)^(3/2 
)*c^3*d^3 + 378*(c*e*x - c*d)^2*sqrt(-c*e*x + c*d)*c^2*d^2 + 180*(c*e*x - 
c*d)^3*sqrt(-c*e*x + c*d)*c*d + 35*(c*e*x - c*d)^4*sqrt(-c*e*x + c*d))*d^2 
/c^3 + 130*(693*sqrt(-c*e*x + c*d)*c^5*d^5 - 1155*(-c*e*x + c*d)^(3/2)*c^4 
*d^4 + 1386*(c*e*x - c*d)^2*sqrt(-c*e*x + c*d)*c^3*d^3 + 990*(c*e*x - c*d) 
^3*sqrt(-c*e*x + c*d)*c^2*d^2 + 385*(c*e*x - c*d)^4*sqrt(-c*e*x + c*d)*c*d 
 + 63*(c*e*x - c*d)^5*sqrt(-c*e*x + c*d))*d/c^4 + 15*(3003*sqrt(-c*e*x + c 
*d)*c^6*d^6 - 6006*(-c*e*x + c*d)^(3/2)*c^5*d^5 + 9009*(c*e*x - c*d)^2*sqr 
t(-c*e*x + c*d)*c^4*d^4 + 8580*(c*e*x - c*d)^3*sqrt(-c*e*x + c*d)*c^3*d^3 
+ 5005*(c*e*x - c*d)^4*sqrt(-c*e*x + c*d)*c^2*d^2 + 1638*(c*e*x - c*d)^5*s 
qrt(-c*e*x + c*d)*c*d + 231*(c*e*x - c*d)^6*sqrt(-c*e*x + c*d))/c^5)/e
 

Mupad [B] (verification not implemented)

Time = 7.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.58 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {16384\,c\,d^6\,\sqrt {c\,d^2-c\,e^2\,x^2}}{15015\,e\,\sqrt {d+e\,x}}-\frac {2\,c\,\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}\,\left (1491\,d^5-4157\,d^4\,e\,x-5350\,d^3\,e^2\,x^2-70\,d^2\,e^3\,x^3+2835\,d\,e^4\,x^4+1155\,e^5\,x^5\right )}{15015\,e} \] Input:

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

Output:

- (16384*c*d^6*(c*d^2 - c*e^2*x^2)^(1/2))/(15015*e*(d + e*x)^(1/2)) - (2*c 
*(c*d^2 - c*e^2*x^2)^(1/2)*(d + e*x)^(1/2)*(1491*d^5 + 1155*e^5*x^5 + 2835 
*d*e^4*x^4 - 5350*d^3*e^2*x^2 - 70*d^2*e^3*x^3 - 4157*d^4*e*x))/(15015*e)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.39 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {-e x +d}\, c \left (-1155 e^{6} x^{6}-3990 d \,e^{5} x^{5}-2765 d^{2} e^{4} x^{4}+5420 d^{3} e^{3} x^{3}+9507 d^{4} e^{2} x^{2}+2666 d^{5} e x -9683 d^{6}\right )}{15015 e} \] Input:

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

Output:

(2*sqrt(c)*sqrt(d - e*x)*c*( - 9683*d**6 + 2666*d**5*e*x + 9507*d**4*e**2* 
x**2 + 5420*d**3*e**3*x**3 - 2765*d**2*e**4*x**4 - 3990*d*e**5*x**5 - 1155 
*e**6*x**6))/(15015*e)