Integrand size = 15, antiderivative size = 164 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=-\frac {2 a^5 \left (a x^2+b x^3\right )^{7/2}}{7 b^6 x^7}+\frac {10 a^4 \left (a x^2+b x^3\right )^{9/2}}{9 b^6 x^9}-\frac {20 a^3 \left (a x^2+b x^3\right )^{11/2}}{11 b^6 x^{11}}+\frac {20 a^2 \left (a x^2+b x^3\right )^{13/2}}{13 b^6 x^{13}}-\frac {2 a \left (a x^2+b x^3\right )^{15/2}}{3 b^6 x^{15}}+\frac {2 \left (a x^2+b x^3\right )^{17/2}}{17 b^6 x^{17}} \] Output:
-2/7*a^5*(b*x^3+a*x^2)^(7/2)/b^6/x^7+10/9*a^4*(b*x^3+a*x^2)^(9/2)/b^6/x^9- 20/11*a^3*(b*x^3+a*x^2)^(11/2)/b^6/x^11+20/13*a^2*(b*x^3+a*x^2)^(13/2)/b^6 /x^13-2/3*a*(b*x^3+a*x^2)^(15/2)/b^6/x^15+2/17*(b*x^3+a*x^2)^(17/2)/b^6/x^ 17
Time = 0.01 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.49 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 x (a+b x)^4 \left (-256 a^5+896 a^4 b x-2016 a^3 b^2 x^2+3696 a^2 b^3 x^3-6006 a b^4 x^4+9009 b^5 x^5\right )}{153153 b^6 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[(a*x^2 + b*x^3)^(5/2),x]
Output:
(2*x*(a + b*x)^4*(-256*a^5 + 896*a^4*b*x - 2016*a^3*b^2*x^2 + 3696*a^2*b^3 *x^3 - 6006*a*b^4*x^4 + 9009*b^5*x^5))/(153153*b^6*Sqrt[x^2*(a + b*x)])
Time = 0.66 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1908, 1922, 1922, 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.
| \(\displaystyle \int \left (a x^2+b x^3\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x}dx}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^2}dx}{15 b}\right )}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^3}dx}{13 b}\right )}{15 b}\right )}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^4}dx}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^5}dx}{9 b}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {4 a \left (a x^2+b x^3\right )^{7/2}}{63 b^2 x^7}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\) |
Input:
Int[(a*x^2 + b*x^3)^(5/2),x]
Output:
(2*(a*x^2 + b*x^3)^(7/2))/(17*b*x^2) - (10*a*((2*(a*x^2 + b*x^3)^(7/2))/(1 5*b*x^3) - (8*a*((2*(a*x^2 + b*x^3)^(7/2))/(13*b*x^4) - (6*a*((2*(a*x^2 + b*x^3)^(7/2))/(11*b*x^5) - (4*a*((-4*a*(a*x^2 + b*x^3)^(7/2))/(63*b^2*x^7) + (2*(a*x^2 + b*x^3)^(7/2))/(9*b*x^6)))/(11*b)))/(13*b)))/(15*b)))/(17*b)
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Simp[b*((n*p + n - j + 1)/(a*( j*p + 1))) Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 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*(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])
Time = 0.37 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.08
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7 b}\) | \(13\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (-9009 b^{5} x^{5}+6006 a \,b^{4} x^{4}-3696 a^{2} b^{3} x^{3}+2016 a^{3} b^{2} x^{2}-896 a^{4} b x +256 a^{5}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{153153 b^{6} x^{5}}\) | \(79\) |
| default | \(-\frac {2 \left (b x +a \right ) \left (-9009 b^{5} x^{5}+6006 a \,b^{4} x^{4}-3696 a^{2} b^{3} x^{3}+2016 a^{3} b^{2} x^{2}-896 a^{4} b x +256 a^{5}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{153153 b^{6} x^{5}}\) | \(79\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (-9009 b^{5} x^{5}+6006 a \,b^{4} x^{4}-3696 a^{2} b^{3} x^{3}+2016 a^{3} b^{2} x^{2}-896 a^{4} b x +256 a^{5}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{153153 b^{6} x^{5}}\) | \(79\) |
| risch | \(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (-9009 b^{8} x^{8}-21021 a \,b^{7} x^{7}-12705 a^{2} b^{6} x^{6}-63 a^{3} b^{5} x^{5}+70 a^{4} b^{4} x^{4}-80 a^{5} b^{3} x^{3}+96 a^{6} b^{2} x^{2}-128 a^{7} b x +256 a^{8}\right )}{153153 x \,b^{6}}\) | \(105\) |
| trager | \(-\frac {2 \left (-9009 b^{8} x^{8}-21021 a \,b^{7} x^{7}-12705 a^{2} b^{6} x^{6}-63 a^{3} b^{5} x^{5}+70 a^{4} b^{4} x^{4}-80 a^{5} b^{3} x^{3}+96 a^{6} b^{2} x^{2}-128 a^{7} b x +256 a^{8}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{153153 b^{6} x}\) | \(107\) |
Input:
int((b*x^3+a*x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/7*(b*x+a)^(7/2)/b
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.65 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{8} x^{8} + 21021 \, a b^{7} x^{7} + 12705 \, a^{2} b^{6} x^{6} + 63 \, a^{3} b^{5} x^{5} - 70 \, a^{4} b^{4} x^{4} + 80 \, a^{5} b^{3} x^{3} - 96 \, a^{6} b^{2} x^{2} + 128 \, a^{7} b x - 256 \, a^{8}\right )} \sqrt {b x^{3} + a x^{2}}}{153153 \, b^{6} x} \] Input:
integrate((b*x^3+a*x^2)^(5/2),x, algorithm="fricas")
Output:
2/153153*(9009*b^8*x^8 + 21021*a*b^7*x^7 + 12705*a^2*b^6*x^6 + 63*a^3*b^5* x^5 - 70*a^4*b^4*x^4 + 80*a^5*b^3*x^3 - 96*a^6*b^2*x^2 + 128*a^7*b*x - 256 *a^8)*sqrt(b*x^3 + a*x^2)/(b^6*x)
\[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=\int \left (a x^{2} + b x^{3}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((b*x**3+a*x**2)**(5/2),x)
Output:
Integral((a*x**2 + b*x**3)**(5/2), x)
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.59 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{8} x^{8} + 21021 \, a b^{7} x^{7} + 12705 \, a^{2} b^{6} x^{6} + 63 \, a^{3} b^{5} x^{5} - 70 \, a^{4} b^{4} x^{4} + 80 \, a^{5} b^{3} x^{3} - 96 \, a^{6} b^{2} x^{2} + 128 \, a^{7} b x - 256 \, a^{8}\right )} \sqrt {b x + a}}{153153 \, b^{6}} \] Input:
integrate((b*x^3+a*x^2)^(5/2),x, algorithm="maxima")
Output:
2/153153*(9009*b^8*x^8 + 21021*a*b^7*x^7 + 12705*a^2*b^6*x^6 + 63*a^3*b^5* x^5 - 70*a^4*b^4*x^4 + 80*a^5*b^3*x^3 - 96*a^6*b^2*x^2 + 128*a^7*b*x - 256 *a^8)*sqrt(b*x + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (140) = 280\).
Time = 0.12 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.41 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {512 \, a^{\frac {17}{2}} \mathrm {sgn}\left (x\right )}{153153 \, b^{6}} + \frac {2 \, {\left (\frac {1105 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a^{3} \mathrm {sgn}\left (x\right )}{b^{5}} + \frac {765 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} a^{2} \mathrm {sgn}\left (x\right )}{b^{5}} + \frac {357 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} a \mathrm {sgn}\left (x\right )}{b^{5}} + \frac {7 \, {\left (6435 \, {\left (b x + a\right )}^{\frac {17}{2}} - 58344 \, {\left (b x + a\right )}^{\frac {15}{2}} a + 235620 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{2} - 556920 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{3} + 850850 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{4} - 875160 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{5} + 612612 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} - 291720 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{7} + 109395 \, \sqrt {b x + a} a^{8}\right )} \mathrm {sgn}\left (x\right )}{b^{5}}\right )}}{765765 \, b} \] Input:
integrate((b*x^3+a*x^2)^(5/2),x, algorithm="giac")
Output:
512/153153*a^(17/2)*sgn(x)/b^6 + 2/765765*(1105*(63*(b*x + a)^(11/2) - 385 *(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*a^3*sgn(x)/b^5 + 765*(23 1*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2) *a^5 + 3003*sqrt(b*x + a)*a^6)*a^2*sgn(x)/b^5 + 357*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^( 9/2)*a^3 + 32175*(b*x + a)^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*( b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*a*sgn(x)/b^5 + 7*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 5569 20*(b*x + a)^(11/2)*a^3 + 850850*(b*x + a)^(9/2)*a^4 - 875160*(b*x + a)^(7 /2)*a^5 + 612612*(b*x + a)^(5/2)*a^6 - 291720*(b*x + a)^(3/2)*a^7 + 109395 *sqrt(b*x + a)*a^8)*sgn(x)/b^5)/b
Time = 8.82 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.49 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=-\frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^3\,\left (256\,a^5-896\,a^4\,b\,x+2016\,a^3\,b^2\,x^2-3696\,a^2\,b^3\,x^3+6006\,a\,b^4\,x^4-9009\,b^5\,x^5\right )}{153153\,b^6\,x} \] Input:
int((a*x^2 + b*x^3)^(5/2),x)
Output:
-(2*(a*x^2 + b*x^3)^(1/2)*(a + b*x)^3*(256*a^5 - 9009*b^5*x^5 + 6006*a*b^4 *x^4 + 2016*a^3*b^2*x^2 - 3696*a^2*b^3*x^3 - 896*a^4*b*x))/(153153*b^6*x)
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.59 \[ \int \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (9009 b^{8} x^{8}+21021 a \,b^{7} x^{7}+12705 a^{2} b^{6} x^{6}+63 a^{3} b^{5} x^{5}-70 a^{4} b^{4} x^{4}+80 a^{5} b^{3} x^{3}-96 a^{6} b^{2} x^{2}+128 a^{7} b x -256 a^{8}\right )}{153153 b^{6}} \] Input:
int((b*x^3+a*x^2)^(5/2),x)
Output:
(2*sqrt(a + b*x)*( - 256*a**8 + 128*a**7*b*x - 96*a**6*b**2*x**2 + 80*a**5 *b**3*x**3 - 70*a**4*b**4*x**4 + 63*a**3*b**5*x**5 + 12705*a**2*b**6*x**6 + 21021*a*b**7*x**7 + 9009*b**8*x**8))/(153153*b**6)