3.2173 \(\int (a+b \sqrt {x})^{15} x \, dx\)

Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4} \]

[Out]

-1/8*a^3*(a+b*x^(1/2))^16/b^4+6/17*a^2*(a+b*x^(1/2))^17/b^4-1/3*a*(a+b*x^(1/2))^18/b^4+2/19*(a+b*x^(1/2))^19/b
^4

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Rubi [A]  time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}-\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15*x,x]

[Out]

-(a^3*(a + b*Sqrt[x])^16)/(8*b^4) + (6*a^2*(a + b*Sqrt[x])^17)/(17*b^4) - (a*(a + b*Sqrt[x])^18)/(3*b^4) + (2*
(a + b*Sqrt[x])^19)/(19*b^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \left (a+b \sqrt {x}\right )^{15} x \, dx &=2 \operatorname {Subst}\left (\int x^3 (a+b x)^{15} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{15}}{b^3}+\frac {3 a^2 (a+b x)^{16}}{b^3}-\frac {3 a (a+b x)^{17}}{b^3}+\frac {(a+b x)^{18}}{b^3}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 50, normalized size = 0.62 \[ -\frac {\left (a+b \sqrt {x}\right )^{16} \left (a^3-16 a^2 b \sqrt {x}+136 a b^2 x-816 b^3 x^{3/2}\right )}{7752 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15*x,x]

[Out]

-1/7752*((a + b*Sqrt[x])^16*(a^3 - 16*a^2*b*Sqrt[x] + 136*a*b^2*x - 816*b^3*x^(3/2)))/b^4

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fricas [B]  time = 1.14, size = 173, normalized size = 2.16 \[ \frac {5}{3} \, a b^{14} x^{9} + \frac {455}{8} \, a^{3} b^{12} x^{8} + 429 \, a^{5} b^{10} x^{7} + \frac {2145}{2} \, a^{7} b^{8} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac {1365}{4} \, a^{11} b^{4} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {1}{2} \, a^{15} x^{2} + \frac {2}{969} \, {\left (51 \, b^{15} x^{9} + 5985 \, a^{2} b^{13} x^{8} + 88179 \, a^{4} b^{11} x^{7} + 373065 \, a^{6} b^{9} x^{6} + 566865 \, a^{8} b^{7} x^{5} + 323323 \, a^{10} b^{5} x^{4} + 62985 \, a^{12} b^{3} x^{3} + 2907 \, a^{14} b x^{2}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^15,x, algorithm="fricas")

[Out]

5/3*a*b^14*x^9 + 455/8*a^3*b^12*x^8 + 429*a^5*b^10*x^7 + 2145/2*a^7*b^8*x^6 + 1001*a^9*b^6*x^5 + 1365/4*a^11*b
^4*x^4 + 35*a^13*b^2*x^3 + 1/2*a^15*x^2 + 2/969*(51*b^15*x^9 + 5985*a^2*b^13*x^8 + 88179*a^4*b^11*x^7 + 373065
*a^6*b^9*x^6 + 566865*a^8*b^7*x^5 + 323323*a^10*b^5*x^4 + 62985*a^12*b^3*x^3 + 2907*a^14*b*x^2)*sqrt(x)

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giac [B]  time = 0.16, size = 167, normalized size = 2.09 \[ \frac {2}{19} \, b^{15} x^{\frac {19}{2}} + \frac {5}{3} \, a b^{14} x^{9} + \frac {210}{17} \, a^{2} b^{13} x^{\frac {17}{2}} + \frac {455}{8} \, a^{3} b^{12} x^{8} + 182 \, a^{4} b^{11} x^{\frac {15}{2}} + 429 \, a^{5} b^{10} x^{7} + 770 \, a^{6} b^{9} x^{\frac {13}{2}} + \frac {2145}{2} \, a^{7} b^{8} x^{6} + 1170 \, a^{8} b^{7} x^{\frac {11}{2}} + 1001 \, a^{9} b^{6} x^{5} + \frac {2002}{3} \, a^{10} b^{5} x^{\frac {9}{2}} + \frac {1365}{4} \, a^{11} b^{4} x^{4} + 130 \, a^{12} b^{3} x^{\frac {7}{2}} + 35 \, a^{13} b^{2} x^{3} + 6 \, a^{14} b x^{\frac {5}{2}} + \frac {1}{2} \, a^{15} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^15,x, algorithm="giac")

[Out]

2/19*b^15*x^(19/2) + 5/3*a*b^14*x^9 + 210/17*a^2*b^13*x^(17/2) + 455/8*a^3*b^12*x^8 + 182*a^4*b^11*x^(15/2) +
429*a^5*b^10*x^7 + 770*a^6*b^9*x^(13/2) + 2145/2*a^7*b^8*x^6 + 1170*a^8*b^7*x^(11/2) + 1001*a^9*b^6*x^5 + 2002
/3*a^10*b^5*x^(9/2) + 1365/4*a^11*b^4*x^4 + 130*a^12*b^3*x^(7/2) + 35*a^13*b^2*x^3 + 6*a^14*b*x^(5/2) + 1/2*a^
15*x^2

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maple [B]  time = 0.00, size = 168, normalized size = 2.10 \[ \frac {2 b^{15} x^{\frac {19}{2}}}{19}+\frac {5 a \,b^{14} x^{9}}{3}+\frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17}+\frac {455 a^{3} b^{12} x^{8}}{8}+182 a^{4} b^{11} x^{\frac {15}{2}}+429 a^{5} b^{10} x^{7}+770 a^{6} b^{9} x^{\frac {13}{2}}+\frac {2145 a^{7} b^{8} x^{6}}{2}+1170 a^{8} b^{7} x^{\frac {11}{2}}+1001 a^{9} b^{6} x^{5}+\frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{4}}{4}+130 a^{12} b^{3} x^{\frac {7}{2}}+35 a^{13} b^{2} x^{3}+6 a^{14} b \,x^{\frac {5}{2}}+\frac {a^{15} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*x^(1/2))^15,x)

[Out]

2/19*x^(19/2)*b^15+5/3*x^9*a*b^14+210/17*x^(17/2)*a^2*b^13+455/8*x^8*a^3*b^12+182*x^(15/2)*a^4*b^11+429*x^7*a^
5*b^10+770*x^(13/2)*a^6*b^9+2145/2*x^6*a^7*b^8+1170*x^(11/2)*a^8*b^7+1001*x^5*a^9*b^6+2002/3*x^(9/2)*a^10*b^5+
1365/4*x^4*a^11*b^4+130*x^(7/2)*a^12*b^3+35*x^3*a^13*b^2+6*x^(5/2)*a^14*b+1/2*x^2*a^15

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maxima [A]  time = 0.89, size = 64, normalized size = 0.80 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{19}}{19 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{18} a}{3 \, b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{17} a^{2}}{17 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{3}}{8 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^15,x, algorithm="maxima")

[Out]

2/19*(b*sqrt(x) + a)^19/b^4 - 1/3*(b*sqrt(x) + a)^18*a/b^4 + 6/17*(b*sqrt(x) + a)^17*a^2/b^4 - 1/8*(b*sqrt(x)
+ a)^16*a^3/b^4

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mupad [B]  time = 0.16, size = 167, normalized size = 2.09 \[ \frac {a^{15}\,x^2}{2}+\frac {2\,b^{15}\,x^{19/2}}{19}+6\,a^{14}\,b\,x^{5/2}+\frac {5\,a\,b^{14}\,x^9}{3}+35\,a^{13}\,b^2\,x^3+\frac {1365\,a^{11}\,b^4\,x^4}{4}+1001\,a^9\,b^6\,x^5+\frac {2145\,a^7\,b^8\,x^6}{2}+429\,a^5\,b^{10}\,x^7+\frac {455\,a^3\,b^{12}\,x^8}{8}+130\,a^{12}\,b^3\,x^{7/2}+\frac {2002\,a^{10}\,b^5\,x^{9/2}}{3}+1170\,a^8\,b^7\,x^{11/2}+770\,a^6\,b^9\,x^{13/2}+182\,a^4\,b^{11}\,x^{15/2}+\frac {210\,a^2\,b^{13}\,x^{17/2}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^(1/2))^15,x)

[Out]

(a^15*x^2)/2 + (2*b^15*x^(19/2))/19 + 6*a^14*b*x^(5/2) + (5*a*b^14*x^9)/3 + 35*a^13*b^2*x^3 + (1365*a^11*b^4*x
^4)/4 + 1001*a^9*b^6*x^5 + (2145*a^7*b^8*x^6)/2 + 429*a^5*b^10*x^7 + (455*a^3*b^12*x^8)/8 + 130*a^12*b^3*x^(7/
2) + (2002*a^10*b^5*x^(9/2))/3 + 1170*a^8*b^7*x^(11/2) + 770*a^6*b^9*x^(13/2) + 182*a^4*b^11*x^(15/2) + (210*a
^2*b^13*x^(17/2))/17

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sympy [B]  time = 5.79, size = 204, normalized size = 2.55 \[ \frac {a^{15} x^{2}}{2} + 6 a^{14} b x^{\frac {5}{2}} + 35 a^{13} b^{2} x^{3} + 130 a^{12} b^{3} x^{\frac {7}{2}} + \frac {1365 a^{11} b^{4} x^{4}}{4} + \frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3} + 1001 a^{9} b^{6} x^{5} + 1170 a^{8} b^{7} x^{\frac {11}{2}} + \frac {2145 a^{7} b^{8} x^{6}}{2} + 770 a^{6} b^{9} x^{\frac {13}{2}} + 429 a^{5} b^{10} x^{7} + 182 a^{4} b^{11} x^{\frac {15}{2}} + \frac {455 a^{3} b^{12} x^{8}}{8} + \frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17} + \frac {5 a b^{14} x^{9}}{3} + \frac {2 b^{15} x^{\frac {19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x**(1/2))**15,x)

[Out]

a**15*x**2/2 + 6*a**14*b*x**(5/2) + 35*a**13*b**2*x**3 + 130*a**12*b**3*x**(7/2) + 1365*a**11*b**4*x**4/4 + 20
02*a**10*b**5*x**(9/2)/3 + 1001*a**9*b**6*x**5 + 1170*a**8*b**7*x**(11/2) + 2145*a**7*b**8*x**6/2 + 770*a**6*b
**9*x**(13/2) + 429*a**5*b**10*x**7 + 182*a**4*b**11*x**(15/2) + 455*a**3*b**12*x**8/8 + 210*a**2*b**13*x**(17
/2)/17 + 5*a*b**14*x**9/3 + 2*b**15*x**(19/2)/19

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