Integrand size = 15, antiderivative size = 220 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=-\frac {2 a^7 \left (a x^2+b x^3\right )^{9/2}}{9 b^8 x^9}+\frac {14 a^6 \left (a x^2+b x^3\right )^{11/2}}{11 b^8 x^{11}}-\frac {42 a^5 \left (a x^2+b x^3\right )^{13/2}}{13 b^8 x^{13}}+\frac {14 a^4 \left (a x^2+b x^3\right )^{15/2}}{3 b^8 x^{15}}-\frac {70 a^3 \left (a x^2+b x^3\right )^{17/2}}{17 b^8 x^{17}}+\frac {42 a^2 \left (a x^2+b x^3\right )^{19/2}}{19 b^8 x^{19}}-\frac {2 a \left (a x^2+b x^3\right )^{21/2}}{3 b^8 x^{21}}+\frac {2 \left (a x^2+b x^3\right )^{23/2}}{23 b^8 x^{23}} \] Output:
-2/9*a^7*(b*x^3+a*x^2)^(9/2)/b^8/x^9+14/11*a^6*(b*x^3+a*x^2)^(11/2)/b^8/x^ 11-42/13*a^5*(b*x^3+a*x^2)^(13/2)/b^8/x^13+14/3*a^4*(b*x^3+a*x^2)^(15/2)/b ^8/x^15-70/17*a^3*(b*x^3+a*x^2)^(17/2)/b^8/x^17+42/19*a^2*(b*x^3+a*x^2)^(1 9/2)/b^8/x^19-2/3*a*(b*x^3+a*x^2)^(21/2)/b^8/x^21+2/23*(b*x^3+a*x^2)^(23/2 )/b^8/x^23
Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.46 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=\frac {2 x (a+b x)^5 \left (-2048 a^7+9216 a^6 b x-25344 a^5 b^2 x^2+54912 a^4 b^3 x^3-102960 a^3 b^4 x^4+175032 a^2 b^5 x^5-277134 a b^6 x^6+415701 b^7 x^7\right )}{9561123 b^8 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[(a*x^2 + b*x^3)^(7/2),x]
Output:
(2*x*(a + b*x)^5*(-2048*a^7 + 9216*a^6*b*x - 25344*a^5*b^2*x^2 + 54912*a^4 *b^3*x^3 - 102960*a^3*b^4*x^4 + 175032*a^2*b^5*x^5 - 277134*a*b^6*x^6 + 41 5701*b^7*x^7))/(9561123*b^8*Sqrt[x^2*(a + b*x)])
Time = 0.85 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1908, 1922, 1922, 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.
| \(\displaystyle \int \left (a x^2+b x^3\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x}dx}{23 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x^2}dx}{7 b}\right )}{23 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{19 b x^4}-\frac {10 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x^3}dx}{19 b}\right )}{7 b}\right )}{23 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{19 b x^4}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{17 b x^5}-\frac {8 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x^4}dx}{17 b}\right )}{19 b}\right )}{7 b}\right )}{23 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{19 b x^4}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{17 b x^5}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{15 b x^6}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x^5}dx}{5 b}\right )}{17 b}\right )}{19 b}\right )}{7 b}\right )}{23 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{19 b x^4}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{17 b x^5}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{15 b x^6}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{13 b x^7}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x^6}dx}{13 b}\right )}{5 b}\right )}{17 b}\right )}{19 b}\right )}{7 b}\right )}{23 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{19 b x^4}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{17 b x^5}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{15 b x^6}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{13 b x^7}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{11 b x^8}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{7/2}}{x^7}dx}{11 b}\right )}{13 b}\right )}{5 b}\right )}{17 b}\right )}{19 b}\right )}{7 b}\right )}{23 b}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 \left (a x^2+b x^3\right )^{9/2}}{23 b x^2}-\frac {14 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{21 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{19 b x^4}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{17 b x^5}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{15 b x^6}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{13 b x^7}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{9/2}}{11 b x^8}-\frac {4 a \left (a x^2+b x^3\right )^{9/2}}{99 b^2 x^9}\right )}{13 b}\right )}{5 b}\right )}{17 b}\right )}{19 b}\right )}{7 b}\right )}{23 b}\) |
Input:
Int[(a*x^2 + b*x^3)^(7/2),x]
Output:
(2*(a*x^2 + b*x^3)^(9/2))/(23*b*x^2) - (14*a*((2*(a*x^2 + b*x^3)^(9/2))/(2 1*b*x^3) - (4*a*((2*(a*x^2 + b*x^3)^(9/2))/(19*b*x^4) - (10*a*((2*(a*x^2 + b*x^3)^(9/2))/(17*b*x^5) - (8*a*((2*(a*x^2 + b*x^3)^(9/2))/(15*b*x^6) - ( 2*a*((2*(a*x^2 + b*x^3)^(9/2))/(13*b*x^7) - (4*a*((-4*a*(a*x^2 + b*x^3)^(9 /2))/(99*b^2*x^9) + (2*(a*x^2 + b*x^3)^(9/2))/(11*b*x^8)))/(13*b)))/(5*b)) )/(17*b)))/(19*b)))/(7*b)))/(23*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 = 0.38 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.06
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9 b}\) | \(13\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (-415701 x^{7} b^{7}+277134 b^{6} a \,x^{6}-175032 a^{2} x^{5} b^{5}+102960 b^{4} x^{4} a^{3}-54912 b^{3} x^{3} a^{4}+25344 x^{2} b^{2} a^{5}-9216 x b \,a^{6}+2048 a^{7}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {7}{2}}}{9561123 b^{8} x^{7}}\) | \(101\) |
| default | \(-\frac {2 \left (b x +a \right ) \left (-415701 x^{7} b^{7}+277134 b^{6} a \,x^{6}-175032 a^{2} x^{5} b^{5}+102960 b^{4} x^{4} a^{3}-54912 b^{3} x^{3} a^{4}+25344 x^{2} b^{2} a^{5}-9216 x b \,a^{6}+2048 a^{7}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {7}{2}}}{9561123 b^{8} x^{7}}\) | \(101\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (-415701 x^{7} b^{7}+277134 b^{6} a \,x^{6}-175032 a^{2} x^{5} b^{5}+102960 b^{4} x^{4} a^{3}-54912 b^{3} x^{3} a^{4}+25344 x^{2} b^{2} a^{5}-9216 x b \,a^{6}+2048 a^{7}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {7}{2}}}{9561123 b^{8} x^{7}}\) | \(101\) |
| risch | \(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (-415701 b^{11} x^{11}-1385670 a \,b^{10} x^{10}-1560702 a^{2} b^{9} x^{9}-597168 a^{3} b^{8} x^{8}-429 a^{4} b^{7} x^{7}+462 a^{5} b^{6} x^{6}-504 a^{6} x^{5} b^{5}+560 a^{7} x^{4} b^{4}-640 a^{8} x^{3} b^{3}+768 a^{9} x^{2} b^{2}-1024 a^{10} b x +2048 a^{11}\right )}{9561123 x \,b^{8}}\) | \(138\) |
| trager | \(-\frac {2 \left (-415701 b^{11} x^{11}-1385670 a \,b^{10} x^{10}-1560702 a^{2} b^{9} x^{9}-597168 a^{3} b^{8} x^{8}-429 a^{4} b^{7} x^{7}+462 a^{5} b^{6} x^{6}-504 a^{6} x^{5} b^{5}+560 a^{7} x^{4} b^{4}-640 a^{8} x^{3} b^{3}+768 a^{9} x^{2} b^{2}-1024 a^{10} b x +2048 a^{11}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{9561123 b^{8} x}\) | \(140\) |
Input:
int((b*x^3+a*x^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
2/9/b*(b*x+a)^(9/2)
Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.63 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=\frac {2 \, {\left (415701 \, b^{11} x^{11} + 1385670 \, a b^{10} x^{10} + 1560702 \, a^{2} b^{9} x^{9} + 597168 \, a^{3} b^{8} x^{8} + 429 \, a^{4} b^{7} x^{7} - 462 \, a^{5} b^{6} x^{6} + 504 \, a^{6} b^{5} x^{5} - 560 \, a^{7} b^{4} x^{4} + 640 \, a^{8} b^{3} x^{3} - 768 \, a^{9} b^{2} x^{2} + 1024 \, a^{10} b x - 2048 \, a^{11}\right )} \sqrt {b x^{3} + a x^{2}}}{9561123 \, b^{8} x} \] Input:
integrate((b*x^3+a*x^2)^(7/2),x, algorithm="fricas")
Output:
2/9561123*(415701*b^11*x^11 + 1385670*a*b^10*x^10 + 1560702*a^2*b^9*x^9 + 597168*a^3*b^8*x^8 + 429*a^4*b^7*x^7 - 462*a^5*b^6*x^6 + 504*a^6*b^5*x^5 - 560*a^7*b^4*x^4 + 640*a^8*b^3*x^3 - 768*a^9*b^2*x^2 + 1024*a^10*b*x - 204 8*a^11)*sqrt(b*x^3 + a*x^2)/(b^8*x)
\[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=\int \left (a x^{2} + b x^{3}\right )^{\frac {7}{2}}\, dx \] Input:
integrate((b*x**3+a*x**2)**(7/2),x)
Output:
Integral((a*x**2 + b*x**3)**(7/2), x)
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.59 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=\frac {2 \, {\left (415701 \, b^{11} x^{11} + 1385670 \, a b^{10} x^{10} + 1560702 \, a^{2} b^{9} x^{9} + 597168 \, a^{3} b^{8} x^{8} + 429 \, a^{4} b^{7} x^{7} - 462 \, a^{5} b^{6} x^{6} + 504 \, a^{6} b^{5} x^{5} - 560 \, a^{7} b^{4} x^{4} + 640 \, a^{8} b^{3} x^{3} - 768 \, a^{9} b^{2} x^{2} + 1024 \, a^{10} b x - 2048 \, a^{11}\right )} \sqrt {b x + a}}{9561123 \, b^{8}} \] Input:
integrate((b*x^3+a*x^2)^(7/2),x, algorithm="maxima")
Output:
2/9561123*(415701*b^11*x^11 + 1385670*a*b^10*x^10 + 1560702*a^2*b^9*x^9 + 597168*a^3*b^8*x^8 + 429*a^4*b^7*x^7 - 462*a^5*b^6*x^6 + 504*a^6*b^5*x^5 - 560*a^7*b^4*x^4 + 640*a^8*b^3*x^3 - 768*a^9*b^2*x^2 + 1024*a^10*b*x - 204 8*a^11)*sqrt(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (188) = 376\).
Time = 0.12 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.92 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x^3+a*x^2)^(7/2),x, algorithm="giac")
Output:
4096/9561123*a^(23/2)*sgn(x)/b^8 + 2/334639305*(52003*(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^4*sgn(x)/b^7 + 12236*(643 5*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^ 2 - 556920*(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)*a^3*sgn(x)/b^7 + 8694*(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2)*a^4 - 3233230*(b*x + a)^(9/2)*a^ 5 + 2771340*(b*x + a)^(7/2)*a^6 - 1662804*(b*x + a)^(5/2)*a^7 + 692835*(b* x + a)^(3/2)*a^8 - 230945*sqrt(b*x + a)*a^9)*a^2*sgn(x)/b^7 + 1380*(46189* (b*x + a)^(21/2) - 510510*(b*x + a)^(19/2)*a + 2567565*(b*x + a)^(17/2)*a^ 2 - 7759752*(b*x + a)^(15/2)*a^3 + 15668730*(b*x + a)^(13/2)*a^4 - 2222110 8*(b*x + a)^(11/2)*a^5 + 22632610*(b*x + a)^(9/2)*a^6 - 16628040*(b*x + a) ^(7/2)*a^7 + 8729721*(b*x + a)^(5/2)*a^8 - 3233230*(b*x + a)^(3/2)*a^9 + 9 69969*sqrt(b*x + a)*a^10)*a*sgn(x)/b^7 + 165*(88179*(b*x + a)^(23/2) - 106 2347*(b*x + a)^(21/2)*a + 5870865*(b*x + a)^(19/2)*a^2 - 19684665*(b*x + a )^(17/2)*a^3 + 44618574*(b*x + a)^(15/2)*a^4 - 72076158*(b*x + a)^(13/2)*a ^5 + 85180914*(b*x + a)^(11/2)*a^6 - 74364290*(b*x + a)^(9/2)*a^7 + 478...
Time = 9.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.46 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=-\frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^4\,\left (2048\,a^7-9216\,a^6\,b\,x+25344\,a^5\,b^2\,x^2-54912\,a^4\,b^3\,x^3+102960\,a^3\,b^4\,x^4-175032\,a^2\,b^5\,x^5+277134\,a\,b^6\,x^6-415701\,b^7\,x^7\right )}{9561123\,b^8\,x} \] Input:
int((a*x^2 + b*x^3)^(7/2),x)
Output:
-(2*(a*x^2 + b*x^3)^(1/2)*(a + b*x)^4*(2048*a^7 - 415701*b^7*x^7 + 277134* a*b^6*x^6 + 25344*a^5*b^2*x^2 - 54912*a^4*b^3*x^3 + 102960*a^3*b^4*x^4 - 1 75032*a^2*b^5*x^5 - 9216*a^6*b*x))/(9561123*b^8*x)
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.59 \[ \int \left (a x^2+b x^3\right )^{7/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (415701 b^{11} x^{11}+1385670 a \,b^{10} x^{10}+1560702 a^{2} b^{9} x^{9}+597168 a^{3} b^{8} x^{8}+429 a^{4} b^{7} x^{7}-462 a^{5} b^{6} x^{6}+504 a^{6} b^{5} x^{5}-560 a^{7} b^{4} x^{4}+640 a^{8} b^{3} x^{3}-768 a^{9} b^{2} x^{2}+1024 a^{10} b x -2048 a^{11}\right )}{9561123 b^{8}} \] Input:
int((b*x^3+a*x^2)^(7/2),x)
Output:
(2*sqrt(a + b*x)*( - 2048*a**11 + 1024*a**10*b*x - 768*a**9*b**2*x**2 + 64 0*a**8*b**3*x**3 - 560*a**7*b**4*x**4 + 504*a**6*b**5*x**5 - 462*a**5*b**6 *x**6 + 429*a**4*b**7*x**7 + 597168*a**3*b**8*x**8 + 1560702*a**2*b**9*x** 9 + 1385670*a*b**10*x**10 + 415701*b**11*x**11))/(9561123*b**8)