Integrand size = 28, antiderivative size = 156 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{13 a (e x)^{27/2}}+\frac {2 (6 b c-13 a d) \left (a x^2+b x^3\right )^{7/2}}{143 a^2 (e x)^{25/2}}-\frac {8 b (6 b c-13 a d) \left (a x^2+b x^3\right )^{7/2}}{1287 a^3 e (e x)^{23/2}}+\frac {16 b^2 (6 b c-13 a d) \left (a x^2+b x^3\right )^{7/2}}{9009 a^4 e^2 (e x)^{21/2}} \] Output:
-2/13*c*e*(b*x^3+a*x^2)^(7/2)/a/(e*x)^(27/2)+2/143*(-13*a*d+6*b*c)*(b*x^3+ a*x^2)^(7/2)/a^2/(e*x)^(25/2)-8/1287*b*(-13*a*d+6*b*c)*(b*x^3+a*x^2)^(7/2) /a^3/e/(e*x)^(23/2)+16/9009*b^2*(-13*a*d+6*b*c)*(b*x^3+a*x^2)^(7/2)/a^4/e^ 2/(e*x)^(21/2)
Time = 0.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=-\frac {2 x (a+b x) \left (x^2 (a+b x)\right )^{5/2} \left (693 a^3 c-378 a^2 b c x+819 a^3 d x+168 a b^2 c x^2-364 a^2 b d x^2-48 b^3 c x^3+104 a b^2 d x^3\right )}{9009 a^4 (e x)^{25/2}} \] Input:
Integrate[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/(e*x)^(25/2),x]
Output:
(-2*x*(a + b*x)*(x^2*(a + b*x))^(5/2)*(693*a^3*c - 378*a^2*b*c*x + 819*a^3 *d*x + 168*a*b^2*c*x^2 - 364*a^2*b*d*x^2 - 48*b^3*c*x^3 + 104*a*b^2*d*x^3) )/(9009*a^4*(e*x)^(25/2))
Time = 0.57 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1944, 1922, 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.
| \(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{5/2} (c+d x)}{(e x)^{25/2}} \, dx\) |
| \(\Big \downarrow \) 1944 |
| \(\displaystyle -\frac {(6 b c-13 a d) \int \frac {\left (b x^3+a x^2\right )^{5/2}}{(e x)^{23/2}}dx}{13 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{13 a (e x)^{27/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {(6 b c-13 a d) \left (-\frac {4 b \int \frac {\left (b x^3+a x^2\right )^{5/2}}{(e x)^{21/2}}dx}{11 a e}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}\right )}{13 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{13 a (e x)^{27/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {(6 b c-13 a d) \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^3+a x^2\right )^{5/2}}{(e x)^{19/2}}dx}{9 a e}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{9 a (e x)^{23/2}}\right )}{11 a e}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}\right )}{13 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{13 a (e x)^{27/2}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle -\frac {(6 b c-13 a d) \left (-\frac {4 b \left (\frac {4 b \left (a x^2+b x^3\right )^{7/2}}{63 a^2 (e x)^{21/2}}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{9 a (e x)^{23/2}}\right )}{11 a e}-\frac {2 e \left (a x^2+b x^3\right )^{7/2}}{11 a (e x)^{25/2}}\right )}{13 a e}-\frac {2 c e \left (a x^2+b x^3\right )^{7/2}}{13 a (e x)^{27/2}}\) |
Input:
Int[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/(e*x)^(25/2),x]
Output:
(-2*c*e*(a*x^2 + b*x^3)^(7/2))/(13*a*(e*x)^(27/2)) - ((6*b*c - 13*a*d)*((- 2*e*(a*x^2 + b*x^3)^(7/2))/(11*a*(e*x)^(25/2)) - (4*b*((-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)))/(13*a*e)
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])
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])
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]
Time = 0.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.58
| method | result | size |
| gosper | \(-\frac {2 x \left (b x +a \right ) \left (104 a \,b^{2} d \,x^{3}-48 b^{3} c \,x^{3}-364 a^{2} b d \,x^{2}+168 a \,b^{2} c \,x^{2}+819 a^{3} d x -378 a^{2} b c x +693 c \,a^{3}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{9009 a^{4} \left (e x \right )^{\frac {25}{2}}}\) | \(91\) |
| orering | \(-\frac {2 x \left (b x +a \right ) \left (104 a \,b^{2} d \,x^{3}-48 b^{3} c \,x^{3}-364 a^{2} b d \,x^{2}+168 a \,b^{2} c \,x^{2}+819 a^{3} d x -378 a^{2} b c x +693 c \,a^{3}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{9009 a^{4} \left (e x \right )^{\frac {25}{2}}}\) | \(91\) |
| default | \(-\frac {2 \left (104 x^{4} a \,b^{3} d -48 x^{4} b^{4} c -260 a^{2} b^{2} d \,x^{3}+120 a \,b^{3} c \,x^{3}+455 a^{3} b d \,x^{2}-210 a^{2} b^{2} c \,x^{2}+819 a^{4} d x +315 a^{3} b c x +693 c \,a^{4}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{9009 \sqrt {e x}\, e^{12} a^{4} x^{11}}\) | \(115\) |
| risch | \(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (104 a \,b^{5} d \,x^{6}-48 b^{6} c \,x^{6}-52 a^{2} b^{4} d \,x^{5}+24 a \,b^{5} c \,x^{5}+39 a^{3} b^{3} d \,x^{4}-18 a^{2} b^{4} c \,x^{4}+1469 a^{4} b^{2} d \,x^{3}+15 a^{3} b^{3} c \,x^{3}+2093 a^{5} b d \,x^{2}+1113 a^{4} b^{2} c \,x^{2}+819 a^{6} d x +1701 a^{5} b c x +693 a^{6} c \right )}{9009 e^{12} x^{7} \sqrt {e x}\, a^{4}}\) | \(161\) |
Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(25/2),x,method=_RETURNVERBOSE)
Output:
-2/9009*x*(b*x+a)*(104*a*b^2*d*x^3-48*b^3*c*x^3-364*a^2*b*d*x^2+168*a*b^2* c*x^2+819*a^3*d*x-378*a^2*b*c*x+693*a^3*c)*(b*x^3+a*x^2)^(5/2)/a^4/(e*x)^( 25/2)
Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=-\frac {2 \, {\left (693 \, a^{6} c - 8 \, {\left (6 \, b^{6} c - 13 \, a b^{5} d\right )} x^{6} + 4 \, {\left (6 \, a b^{5} c - 13 \, a^{2} b^{4} d\right )} x^{5} - 3 \, {\left (6 \, a^{2} b^{4} c - 13 \, a^{3} b^{3} d\right )} x^{4} + {\left (15 \, a^{3} b^{3} c + 1469 \, a^{4} b^{2} d\right )} x^{3} + 7 \, {\left (159 \, a^{4} b^{2} c + 299 \, a^{5} b d\right )} x^{2} + 63 \, {\left (27 \, a^{5} b c + 13 \, a^{6} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}} \sqrt {e x}}{9009 \, a^{4} e^{13} x^{8}} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(25/2),x, algorithm="fricas")
Output:
-2/9009*(693*a^6*c - 8*(6*b^6*c - 13*a*b^5*d)*x^6 + 4*(6*a*b^5*c - 13*a^2* b^4*d)*x^5 - 3*(6*a^2*b^4*c - 13*a^3*b^3*d)*x^4 + (15*a^3*b^3*c + 1469*a^4 *b^2*d)*x^3 + 7*(159*a^4*b^2*c + 299*a^5*b*d)*x^2 + 63*(27*a^5*b*c + 13*a^ 6*d)*x)*sqrt(b*x^3 + a*x^2)*sqrt(e*x)/(a^4*e^13*x^8)
Timed out. \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)*(b*x**3+a*x**2)**(5/2)/(e*x)**(25/2),x)
Output:
Timed out
\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}} {\left (d x + c\right )}}{\left (e x\right )^{\frac {25}{2}}} \,d x } \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(25/2),x, algorithm="maxima")
Output:
integrate((b*x^3 + a*x^2)^(5/2)*(d*x + c)/(e*x)^(25/2), x)
Time = 0.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (6 \, a^{2} b^{13} c e^{6} \mathrm {sgn}\left (x\right ) - 13 \, a^{3} b^{12} d e^{6} \mathrm {sgn}\left (x\right )\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (6 \, a^{3} b^{13} c e^{6} \mathrm {sgn}\left (x\right ) - 13 \, a^{4} b^{12} d e^{6} \mathrm {sgn}\left (x\right )\right )}}{a^{6}}\right )} + \frac {143 \, {\left (6 \, a^{4} b^{13} c e^{6} \mathrm {sgn}\left (x\right ) - 13 \, a^{5} b^{12} d e^{6} \mathrm {sgn}\left (x\right )\right )}}{a^{6}}\right )} - \frac {1287 \, {\left (a^{5} b^{13} c e^{6} \mathrm {sgn}\left (x\right ) - a^{6} b^{12} d e^{6} \mathrm {sgn}\left (x\right )\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{9009 \, {\left ({\left (b x + a\right )} b e - a b e\right )}^{\frac {13}{2}} e^{12} {\left | b \right |}} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(25/2),x, algorithm="giac")
Output:
2/9009*((b*x + a)*(4*(b*x + a)*(2*(6*a^2*b^13*c*e^6*sgn(x) - 13*a^3*b^12*d *e^6*sgn(x))*(b*x + a)/a^6 - 13*(6*a^3*b^13*c*e^6*sgn(x) - 13*a^4*b^12*d*e ^6*sgn(x))/a^6) + 143*(6*a^4*b^13*c*e^6*sgn(x) - 13*a^5*b^12*d*e^6*sgn(x)) /a^6) - 1287*(a^5*b^13*c*e^6*sgn(x) - a^6*b^12*d*e^6*sgn(x))/a^6)*(b*x + a )^(7/2)*b/(((b*x + a)*b*e - a*b*e)^(13/2)*e^12*abs(b))
Time = 9.68 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=-\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {2\,a^2\,c}{13\,e^{12}}+\frac {2\,b\,x^2\,\left (299\,a\,d+159\,b\,c\right )}{1287\,e^{12}}+\frac {x\,\left (1638\,d\,a^6+3402\,b\,c\,a^5\right )}{9009\,a^4\,e^{12}}-\frac {x^6\,\left (96\,b^6\,c-208\,a\,b^5\,d\right )}{9009\,a^4\,e^{12}}+\frac {2\,b^3\,x^4\,\left (13\,a\,d-6\,b\,c\right )}{3003\,a^2\,e^{12}}-\frac {8\,b^4\,x^5\,\left (13\,a\,d-6\,b\,c\right )}{9009\,a^3\,e^{12}}+\frac {2\,b^2\,x^3\,\left (1469\,a\,d+15\,b\,c\right )}{9009\,a\,e^{12}}\right )}{x^7\,\sqrt {e\,x}} \] Input:
int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/(e*x)^(25/2),x)
Output:
-((a*x^2 + b*x^3)^(1/2)*((2*a^2*c)/(13*e^12) + (2*b*x^2*(299*a*d + 159*b*c ))/(1287*e^12) + (x*(1638*a^6*d + 3402*a^5*b*c))/(9009*a^4*e^12) - (x^6*(9 6*b^6*c - 208*a*b^5*d))/(9009*a^4*e^12) + (2*b^3*x^4*(13*a*d - 6*b*c))/(30 03*a^2*e^12) - (8*b^4*x^5*(13*a*d - 6*b*c))/(9009*a^3*e^12) + (2*b^2*x^3*( 1469*a*d + 15*b*c))/(9009*a*e^12)))/(x^7*(e*x)^(1/2))
Time = 0.32 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.75 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{(e x)^{25/2}} \, dx=\frac {2 \sqrt {e}\, \left (-693 \sqrt {x}\, \sqrt {b x +a}\, a^{6} c -819 \sqrt {x}\, \sqrt {b x +a}\, a^{6} d x -1701 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b c x -2093 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b d \,x^{2}-1113 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} c \,x^{2}-1469 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} d \,x^{3}-15 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c \,x^{3}-39 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} d \,x^{4}+18 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c \,x^{4}+52 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} d \,x^{5}-24 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c \,x^{5}-104 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} d \,x^{6}+48 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c \,x^{6}+104 \sqrt {b}\, a \,b^{5} d \,x^{7}-48 \sqrt {b}\, b^{6} c \,x^{7}\right )}{9009 a^{4} e^{13} x^{7}} \] Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/(e*x)^(25/2),x)
Output:
(2*sqrt(e)*( - 693*sqrt(x)*sqrt(a + b*x)*a**6*c - 819*sqrt(x)*sqrt(a + b*x )*a**6*d*x - 1701*sqrt(x)*sqrt(a + b*x)*a**5*b*c*x - 2093*sqrt(x)*sqrt(a + b*x)*a**5*b*d*x**2 - 1113*sqrt(x)*sqrt(a + b*x)*a**4*b**2*c*x**2 - 1469*s qrt(x)*sqrt(a + b*x)*a**4*b**2*d*x**3 - 15*sqrt(x)*sqrt(a + b*x)*a**3*b**3 *c*x**3 - 39*sqrt(x)*sqrt(a + b*x)*a**3*b**3*d*x**4 + 18*sqrt(x)*sqrt(a + b*x)*a**2*b**4*c*x**4 + 52*sqrt(x)*sqrt(a + b*x)*a**2*b**4*d*x**5 - 24*sqr t(x)*sqrt(a + b*x)*a*b**5*c*x**5 - 104*sqrt(x)*sqrt(a + b*x)*a*b**5*d*x**6 + 48*sqrt(x)*sqrt(a + b*x)*b**6*c*x**6 + 104*sqrt(b)*a*b**5*d*x**7 - 48*s qrt(b)*b**6*c*x**7))/(9009*a**4*e**13*x**7)