\(\int \frac {(c+d x) (a x^2+b x^3)^{5/2}}{(e x)^{23/2}} \, dx\) [330]

Optimal result

Integrand size = 28, antiderivative size = 112 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}+\frac {2 (4 b c-11 a d) \left (a x^2+b x^3\right )^{7/2}}{99 a^2 (e x)^{23/2}}-\frac {4 b (4 b c-11 a d) \left (a x^2+b x^3\right )^{7/2}}{693 a^3 e (e x)^{21/2}} \] Output:

-2/11*c*e*(b*x^3+a*x^2)^(7/2)/a/(e*x)^(25/2)+2/99*(-11*a*d+4*b*c)*(b*x^3+a 
*x^2)^(7/2)/a^2/(e*x)^(23/2)-4/693*b*(-11*a*d+4*b*c)*(b*x^3+a*x^2)^(7/2)/a 
^3/e/(e*x)^(21/2)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=-\frac {2 e \left (x^2 (a+b x)\right )^{7/2} \left (8 b^2 c x^2+7 a^2 (9 c+11 d x)-2 a b x (14 c+11 d x)\right )}{693 a^3 (e x)^{25/2}} \] Input:

Integrate[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/(e*x)^(23/2),x]
 

Output:

(-2*e*(x^2*(a + b*x))^(7/2)*(8*b^2*c*x^2 + 7*a^2*(9*c + 11*d*x) - 2*a*b*x* 
(14*c + 11*d*x)))/(693*a^3*(e*x)^(25/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1944, 1922, 1920}

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.

\(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{5/2} (c+d x)}{(e x)^{23/2}} \, dx\)

\(\Big \downarrow \) 1944

\(\displaystyle -\frac {(4 b c-11 a d) \int \frac {\left (b x^3+a x^2\right )^{5/2}}{(e x)^{21/2}}dx}{11 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}\)

\(\Big \downarrow \) 1922

\(\displaystyle -\frac {(4 b c-11 a d) \left (-\frac {2 b \int \frac {\left (b x^3+a x^2\right )^{5/2}}{(e x)^{19/2}}dx}{9 a e}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{9 a (e x)^{23/2}}\right )}{11 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}\)

\(\Big \downarrow \) 1920

\(\displaystyle -\frac {(4 b c-11 a d) \left (\frac {4 b \left (a x^2+b x^3\right )^{7/2}}{63 a^2 (e x)^{21/2}}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{9 a (e x)^{23/2}}\right )}{11 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}\)

Input:

Int[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/(e*x)^(23/2),x]
 

Output:

(-2*c*e*(a*x^2 + b*x^3)^(7/2))/(11*a*(e*x)^(25/2)) - ((4*b*c - 11*a*d)*((- 
2*e*(a*x^2 + b*x^3)^(7/2))/(9*a*(e*x)^(23/2)) + (4*b*(a*x^2 + b*x^3)^(7/2) 
)/(63*a^2*(e*x)^(21/2))))/(11*a*e)
 

Defintions of rubi rules used

rule 1920

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

rule 1922

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, 
p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) 
/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
 

rule 1944

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.60

method	result	size
gosper	\(-\frac {2 x \left (b x +a \right ) \left (-22 a b d \,x^{2}+8 b^{2} c \,x^{2}+77 a^{2} d x -28 a b c x +63 a^{2} c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{693 a^{3} \left (e x \right )^{\frac {23}{2}}}\)	\(67\)
orering	\(-\frac {2 x \left (b x +a \right ) \left (-22 a b d \,x^{2}+8 b^{2} c \,x^{2}+77 a^{2} d x -28 a b c x +63 a^{2} c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{693 a^{3} \left (e x \right )^{\frac {23}{2}}}\)	\(67\)
default	\(-\frac {2 \left (-22 a \,b^{2} d \,x^{3}+8 b^{3} c \,x^{3}+55 a^{2} b d \,x^{2}-20 a \,b^{2} c \,x^{2}+77 a^{3} d x +35 a^{2} b c x +63 c \,a^{3}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{693 \sqrt {e x}\, e^{11} a^{3} x^{10}}\)	\(91\)
risch	\(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (-22 a \,b^{4} d \,x^{5}+8 b^{5} c \,x^{5}+11 x^{4} a^{2} b^{3} d -4 x^{4} a \,b^{4} c +165 a^{3} b^{2} d \,x^{3}+3 a^{2} b^{3} c \,x^{3}+209 a^{4} b d \,x^{2}+113 a^{3} b^{2} c \,x^{2}+77 a^{5} d x +161 a^{4} b c x +63 c \,a^{5}\right )}{693 e^{11} x^{6} \sqrt {e x}\, a^{3}}\)	\(137\)

Input:

int((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(23/2),x,method=_RETURNVERBOSE)
 

Output:

-2/693*x*(b*x+a)*(-22*a*b*d*x^2+8*b^2*c*x^2+77*a^2*d*x-28*a*b*c*x+63*a^2*c 
)*(b*x^3+a*x^2)^(5/2)/a^3/(e*x)^(23/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.23 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=-\frac {2 \, {\left (63 \, a^{5} c + 2 \, {\left (4 \, b^{5} c - 11 \, a b^{4} d\right )} x^{5} - {\left (4 \, a b^{4} c - 11 \, a^{2} b^{3} d\right )} x^{4} + 3 \, {\left (a^{2} b^{3} c + 55 \, a^{3} b^{2} d\right )} x^{3} + {\left (113 \, a^{3} b^{2} c + 209 \, a^{4} b d\right )} x^{2} + 7 \, {\left (23 \, a^{4} b c + 11 \, a^{5} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}} \sqrt {e x}}{693 \, a^{3} e^{12} x^{7}} \] Input:

integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(23/2),x, algorithm="fricas")
 

Output:

-2/693*(63*a^5*c + 2*(4*b^5*c - 11*a*b^4*d)*x^5 - (4*a*b^4*c - 11*a^2*b^3* 
d)*x^4 + 3*(a^2*b^3*c + 55*a^3*b^2*d)*x^3 + (113*a^3*b^2*c + 209*a^4*b*d)* 
x^2 + 7*(23*a^4*b*c + 11*a^5*d)*x)*sqrt(b*x^3 + a*x^2)*sqrt(e*x)/(a^3*e^12 
*x^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)*(b*x**3+a*x**2)**(5/2)/(e*x)**(23/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}} {\left (d x + c\right )}}{\left (e x\right )^{\frac {23}{2}}} \,d x } \] Input:

integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(23/2),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a*x^2)^(5/2)*(d*x + c)/(e*x)^(23/2), x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=-\frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (4 \, a^{2} b^{5} c e^{5} \mathrm {sgn}\left (x\right ) - 11 \, a^{3} b^{4} d e^{5} \mathrm {sgn}\left (x\right )\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (4 \, a^{3} b^{5} c e^{5} \mathrm {sgn}\left (x\right ) - 11 \, a^{4} b^{4} d e^{5} \mathrm {sgn}\left (x\right )\right )}}{a^{5}}\right )} + \frac {99 \, {\left (a^{4} b^{5} c e^{5} \mathrm {sgn}\left (x\right ) - a^{5} b^{4} d e^{5} \mathrm {sgn}\left (x\right )\right )}}{a^{5}}\right )} b^{7}}{693 \, {\left ({\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}} e^{11} {\left | b \right |}} \] Input:

integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(23/2),x, algorithm="giac")
 

Output:

-2/693*(b*x + a)^(7/2)*((b*x + a)*(2*(4*a^2*b^5*c*e^5*sgn(x) - 11*a^3*b^4* 
d*e^5*sgn(x))*(b*x + a)/a^5 - 11*(4*a^3*b^5*c*e^5*sgn(x) - 11*a^4*b^4*d*e^ 
5*sgn(x))/a^5) + 99*(a^4*b^5*c*e^5*sgn(x) - a^5*b^4*d*e^5*sgn(x))/a^5)*b^7 
/(((b*x + a)*b*e - a*b*e)^(11/2)*e^11*abs(b))
 

Mupad [B] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.29 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=-\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {2\,a^2\,c}{11\,e^{11}}+\frac {2\,b\,x^2\,\left (209\,a\,d+113\,b\,c\right )}{693\,e^{11}}+\frac {x\,\left (154\,d\,a^5+322\,b\,c\,a^4\right )}{693\,a^3\,e^{11}}+\frac {x^5\,\left (16\,b^5\,c-44\,a\,b^4\,d\right )}{693\,a^3\,e^{11}}+\frac {2\,b^3\,x^4\,\left (11\,a\,d-4\,b\,c\right )}{693\,a^2\,e^{11}}+\frac {2\,b^2\,x^3\,\left (55\,a\,d+b\,c\right )}{231\,a\,e^{11}}\right )}{x^6\,\sqrt {e\,x}} \] Input:

int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/(e*x)^(23/2),x)
 

Output:

-((a*x^2 + b*x^3)^(1/2)*((2*a^2*c)/(11*e^11) + (2*b*x^2*(209*a*d + 113*b*c 
))/(693*e^11) + (x*(154*a^5*d + 322*a^4*b*c))/(693*a^3*e^11) + (x^5*(16*b^ 
5*c - 44*a*b^4*d))/(693*a^3*e^11) + (2*b^3*x^4*(11*a*d - 4*b*c))/(693*a^2* 
e^11) + (2*b^2*x^3*(55*a*d + b*c))/(231*a*e^11)))/(x^6*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{23/2}} \, dx=\frac {2 \sqrt {e}\, \left (-63 \sqrt {x}\, \sqrt {b x +a}\, a^{5} c -77 \sqrt {x}\, \sqrt {b x +a}\, a^{5} d x -161 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b c x -209 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b d \,x^{2}-113 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} c \,x^{2}-165 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} d \,x^{3}-3 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c \,x^{3}-11 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} d \,x^{4}+4 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c \,x^{4}+22 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} d \,x^{5}-8 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c \,x^{5}-22 \sqrt {b}\, a \,b^{4} d \,x^{6}+8 \sqrt {b}\, b^{5} c \,x^{6}\right )}{693 a^{3} e^{12} x^{6}} \] Input:

int((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(23/2),x)
 

Output:

(2*sqrt(e)*( - 63*sqrt(x)*sqrt(a + b*x)*a**5*c - 77*sqrt(x)*sqrt(a + b*x)* 
a**5*d*x - 161*sqrt(x)*sqrt(a + b*x)*a**4*b*c*x - 209*sqrt(x)*sqrt(a + b*x 
)*a**4*b*d*x**2 - 113*sqrt(x)*sqrt(a + b*x)*a**3*b**2*c*x**2 - 165*sqrt(x) 
*sqrt(a + b*x)*a**3*b**2*d*x**3 - 3*sqrt(x)*sqrt(a + b*x)*a**2*b**3*c*x**3 
 - 11*sqrt(x)*sqrt(a + b*x)*a**2*b**3*d*x**4 + 4*sqrt(x)*sqrt(a + b*x)*a*b 
**4*c*x**4 + 22*sqrt(x)*sqrt(a + b*x)*a*b**4*d*x**5 - 8*sqrt(x)*sqrt(a + b 
*x)*b**5*c*x**5 - 22*sqrt(b)*a*b**4*d*x**6 + 8*sqrt(b)*b**5*c*x**6))/(693* 
a**3*e**12*x**6)