Integrand size = 15, antiderivative size = 169 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {512 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}} \] Output:
2/5*(b*x^(2/3)+a*x)^(5/2)/a-512/15015*b^5*(b*x^(2/3)+a*x)^(5/2)/a^6/x^(5/3 )+256/3003*b^4*(b*x^(2/3)+a*x)^(5/2)/a^5/x^(4/3)-64/429*b^3*(b*x^(2/3)+a*x )^(5/2)/a^4/x+32/143*b^2*(b*x^(2/3)+a*x)^(5/2)/a^3/x^(2/3)-4/13*b*(b*x^(2/ 3)+a*x)^(5/2)/a^2/x^(1/3)
Time = 5.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.56 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \left (b+a \sqrt [3]{x}\right ) \left (b x^{2/3}+a x\right )^{3/2} \left (-256 b^5+640 a b^4 \sqrt [3]{x}-1120 a^2 b^3 x^{2/3}+1680 a^3 b^2 x-2310 a^4 b x^{4/3}+3003 a^5 x^{5/3}\right )}{15015 a^6 x} \] Input:
Integrate[(b*x^(2/3) + a*x)^(3/2),x]
Output:
(2*(b + a*x^(1/3))*(b*x^(2/3) + a*x)^(3/2)*(-256*b^5 + 640*a*b^4*x^(1/3) - 1120*a^2*b^3*x^(2/3) + 1680*a^3*b^2*x - 2310*a^4*b*x^(4/3) + 3003*a^5*x^( 5/3)))/(15015*a^6*x)
Time = 0.61 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.14, 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.
\(\displaystyle \int \left (a x+b x^{2/3}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1908 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{\sqrt [3]{x}}dx}{3 a}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{2/3}}dx}{13 a}\right )}{3 a}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x}dx}{11 a}\right )}{13 a}\right )}{3 a}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{4/3}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{7 a x^{4/3}}-\frac {2 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{5/3}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{7 a x^{4/3}}-\frac {12 b \left (a x+b x^{2/3}\right )^{5/2}}{35 a^2 x^{5/3}}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\) |
Input:
Int[(b*x^(2/3) + a*x)^(3/2),x]
Output:
(2*(b*x^(2/3) + a*x)^(5/2))/(5*a) - (2*b*((6*(b*x^(2/3) + a*x)^(5/2))/(13* a*x^(1/3)) - (8*b*((6*(b*x^(2/3) + a*x)^(5/2))/(11*a*x^(2/3)) - (6*b*((2*( b*x^(2/3) + a*x)^(5/2))/(3*a*x) - (4*b*((-12*b*(b*x^(2/3) + a*x)^(5/2))/(3 5*a^2*x^(5/3)) + (6*(b*x^(2/3) + a*x)^(5/2))/(7*a*x^(4/3))))/(9*a)))/(11*a )))/(13*a)))/(3*a)
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 = 79, normalized size of antiderivative = 0.47
method | result | size |
derivativedivides | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (x^{\frac {1}{3}} a +b \right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 x^{\frac {4}{3}} a^{4} b +1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 x^{\frac {1}{3}} a \,b^{4}-256 b^{5}\right )}{15015 x \,a^{6}}\) | \(79\) |
default | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (x^{\frac {1}{3}} a +b \right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 x^{\frac {4}{3}} a^{4} b +1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 x^{\frac {1}{3}} a \,b^{4}-256 b^{5}\right )}{15015 x \,a^{6}}\) | \(79\) |
Input:
int((b*x^(2/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/15015*(b*x^(2/3)+a*x)^(3/2)*(x^(1/3)*a+b)*(3003*a^5*x^(5/3)-2310*x^(4/3) *a^4*b+1680*a^3*b^2*x-1120*a^2*b^3*x^(2/3)+640*x^(1/3)*a*b^4-256*b^5)/x/a^ 6
Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (125) = 250\).
Time = 126.39 (sec) , antiderivative size = 768, normalized size of antiderivative = 4.54 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
2/15015*(4*(805306368*b^13 + 167772160*b^12 + 786432*(64*a^3 - 3)*b^10 - 1 5728640*b^11 - 4096*(11264*a^3 - 53)*b^9 + 4372368*a^9 - 1536*(5504*a^3 + 1)*b^8 - 48*(242810880*a^6 + 114688*a^3 + 3)*b^7 - 1792*(1353984*a^6 - 103 *a^3)*b^6 + 192*(1152384*a^6 - 23*a^3)*b^5 - 3*(3633315840*a^9 - 12027392* a^6 - 15*a^3)*b^4 - 112*(35389440*a^9 + 29281*a^6)*b^3 - 819*(368640*a^9 - 31*a^6)*b^2 + 693*(40960*a^9 + 3*a^6)*b)*x + (3003*(16777216*a^7*b^6 + 62 91456*a^7*b^5 + 196608*a^7*b^4 - 262144*a^10 - 114688*a^7*b^3 - 2304*a^7*b ^2 + 864*a^7*b - 27*a^7)*x^3 - 70*(16777216*a^4*b^9 + 6291456*a^4*b^8 + 19 6608*a^4*b^7 - 114688*a^4*b^6 - 2304*a^4*b^5 + 864*a^4*b^4 - (262144*a^7 + 27*a^4)*b^3)*x^2 + 128*(16777216*a*b^12 + 6291456*a*b^11 + 196608*a*b^10 - 114688*a*b^9 - 2304*a*b^8 + 864*a*b^7 - (262144*a^4 + 27*a)*b^6)*x - 16* (268435456*b^13 + 100663296*b^12 + 3145728*b^11 - 1835008*b^10 - 36864*b^9 - 16*(262144*a^3 + 27)*b^7 + 13824*b^8 - 231*(16777216*a^6*b^7 + 6291456* a^6*b^6 + 196608*a^6*b^5 - 114688*a^6*b^4 - 2304*a^6*b^3 + 864*a^6*b^2 - ( 262144*a^9 + 27*a^6)*b)*x^2 - 5*(16777216*a^3*b^10 + 6291456*a^3*b^9 + 196 608*a^3*b^8 - 114688*a^3*b^7 - 2304*a^3*b^6 + 864*a^3*b^5 - (262144*a^6 + 27*a^3)*b^4)*x)*x^(2/3) + 3*(21*(16777216*a^5*b^8 + 6291456*a^5*b^7 + 1966 08*a^5*b^6 - 114688*a^5*b^5 - 2304*a^5*b^4 + 864*a^5*b^3 - (262144*a^8 + 2 7*a^5)*b^2)*x^2 - 32*(16777216*a^2*b^11 + 6291456*a^2*b^10 + 196608*a^2*b^ 9 - 114688*a^2*b^8 - 2304*a^2*b^7 + 864*a^2*b^6 - (262144*a^5 + 27*a^2)...
\[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((b*x**(2/3)+a*x)**(3/2),x)
Output:
Integral((a*x + b*x**(2/3))**(3/2), x)
\[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((b*x^(2/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate((a*x + b*x^(2/3))^(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (125) = 250\).
Time = 0.28 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.57 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2),x, algorithm="giac")
Output:
2/3003*b*(256*b^(13/2)/a^6 + (13*(63*(a*x^(1/3) + b)^(11/2) - 385*(a*x^(1/ 3) + b)^(9/2)*b + 990*(a*x^(1/3) + b)^(7/2)*b^2 - 1386*(a*x^(1/3) + b)^(5/ 2)*b^3 + 1155*(a*x^(1/3) + b)^(3/2)*b^4 - 693*sqrt(a*x^(1/3) + b)*b^5)*b/a ^5 + 3*(231*(a*x^(1/3) + b)^(13/2) - 1638*(a*x^(1/3) + b)^(11/2)*b + 5005* (a*x^(1/3) + b)^(9/2)*b^2 - 8580*(a*x^(1/3) + b)^(7/2)*b^3 + 9009*(a*x^(1/ 3) + b)^(5/2)*b^4 - 6006*(a*x^(1/3) + b)^(3/2)*b^5 + 3003*sqrt(a*x^(1/3) + b)*b^6)/a^5)/a) - 2/15015*a*(1024*b^(15/2)/a^7 - (15*(231*(a*x^(1/3) + b) ^(13/2) - 1638*(a*x^(1/3) + b)^(11/2)*b + 5005*(a*x^(1/3) + b)^(9/2)*b^2 - 8580*(a*x^(1/3) + b)^(7/2)*b^3 + 9009*(a*x^(1/3) + b)^(5/2)*b^4 - 6006*(a *x^(1/3) + b)^(3/2)*b^5 + 3003*sqrt(a*x^(1/3) + b)*b^6)*b/a^6 + 7*(429*(a* x^(1/3) + b)^(15/2) - 3465*(a*x^(1/3) + b)^(13/2)*b + 12285*(a*x^(1/3) + b )^(11/2)*b^2 - 25025*(a*x^(1/3) + b)^(9/2)*b^3 + 32175*(a*x^(1/3) + b)^(7/ 2)*b^4 - 27027*(a*x^(1/3) + b)^(5/2)*b^5 + 15015*(a*x^(1/3) + b)^(3/2)*b^6 - 6435*sqrt(a*x^(1/3) + b)*b^7)/a^6)/a)
Time = 8.74 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.24 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {x\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},6;\ 7;\ -\frac {a\,x^{1/3}}{b}\right )}{2\,{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{3/2}} \] Input:
int((a*x + b*x^(2/3))^(3/2),x)
Output:
(x*(a*x + b*x^(2/3))^(3/2)*hypergeom([-3/2, 6], 7, -(a*x^(1/3))/b))/(2*((a *x^(1/3))/b + 1)^(3/2))
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.51 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \sqrt {x^{\frac {1}{3}} a +b}\, \left (63 x^{\frac {5}{3}} a^{5} b^{2}-96 x^{\frac {2}{3}} a^{2} b^{5}+3003 x^{\frac {7}{3}} a^{7}-70 x^{\frac {4}{3}} a^{4} b^{3}+128 x^{\frac {1}{3}} a \,b^{6}+3696 a^{6} b \,x^{2}+80 a^{3} b^{4} x -256 b^{7}\right )}{15015 a^{6}} \] Input:
int((b*x^(2/3)+a*x)^(3/2),x)
Output:
(2*sqrt(x**(1/3)*a + b)*(63*x**(2/3)*a**5*b**2*x - 96*x**(2/3)*a**2*b**5 + 3003*x**(1/3)*a**7*x**2 - 70*x**(1/3)*a**4*b**3*x + 128*x**(1/3)*a*b**6 + 3696*a**6*b*x**2 + 80*a**3*b**4*x - 256*b**7))/(15015*a**6)