3.2153 \(\int (a+b \sqrt {x})^{10} x^4 \, dx\)

Optimal. Leaf size=140 \[ \frac {a^{10} x^5}{5}+\frac {20}{11} a^9 b x^{11/2}+\frac {15}{2} a^8 b^2 x^6+\frac {240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac {168}{5} a^5 b^5 x^{15/2}+\frac {105}{4} a^4 b^6 x^8+\frac {240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac {20}{19} a b^9 x^{19/2}+\frac {b^{10} x^{10}}{10} \]

[Out]

1/5*a^10*x^5+20/11*a^9*b*x^(11/2)+15/2*a^8*b^2*x^6+240/13*a^7*b^3*x^(13/2)+30*a^6*b^4*x^7+168/5*a^5*b^5*x^(15/
2)+105/4*a^4*b^6*x^8+240/17*a^3*b^7*x^(17/2)+5*a^2*b^8*x^9+20/19*a*b^9*x^(19/2)+1/10*b^10*x^10

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Rubi [A]  time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {15}{2} a^8 b^2 x^6+\frac {240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac {168}{5} a^5 b^5 x^{15/2}+\frac {105}{4} a^4 b^6 x^8+\frac {240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac {20}{11} a^9 b x^{11/2}+\frac {a^{10} x^5}{5}+\frac {20}{19} a b^9 x^{19/2}+\frac {b^{10} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10*x^4,x]

[Out]

(a^10*x^5)/5 + (20*a^9*b*x^(11/2))/11 + (15*a^8*b^2*x^6)/2 + (240*a^7*b^3*x^(13/2))/13 + 30*a^6*b^4*x^7 + (168
*a^5*b^5*x^(15/2))/5 + (105*a^4*b^6*x^8)/4 + (240*a^3*b^7*x^(17/2))/17 + 5*a^2*b^8*x^9 + (20*a*b^9*x^(19/2))/1
9 + (b^10*x^10)/10

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 )^{10} x^4 \, dx &=2 \operatorname {Subst}\left (\int x^9 (a+b x)^{10} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^{10} x^9+10 a^9 b x^{10}+45 a^8 b^2 x^{11}+120 a^7 b^3 x^{12}+210 a^6 b^4 x^{13}+252 a^5 b^5 x^{14}+210 a^4 b^6 x^{15}+120 a^3 b^7 x^{16}+45 a^2 b^8 x^{17}+10 a b^9 x^{18}+b^{10} x^{19}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^{10} x^5}{5}+\frac {20}{11} a^9 b x^{11/2}+\frac {15}{2} a^8 b^2 x^6+\frac {240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac {168}{5} a^5 b^5 x^{15/2}+\frac {105}{4} a^4 b^6 x^8+\frac {240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac {20}{19} a b^9 x^{19/2}+\frac {b^{10} x^{10}}{10}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 140, normalized size = 1.00 \[ \frac {a^{10} x^5}{5}+\frac {20}{11} a^9 b x^{11/2}+\frac {15}{2} a^8 b^2 x^6+\frac {240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac {168}{5} a^5 b^5 x^{15/2}+\frac {105}{4} a^4 b^6 x^8+\frac {240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac {20}{19} a b^9 x^{19/2}+\frac {b^{10} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10*x^4,x]

[Out]

(a^10*x^5)/5 + (20*a^9*b*x^(11/2))/11 + (15*a^8*b^2*x^6)/2 + (240*a^7*b^3*x^(13/2))/13 + 30*a^6*b^4*x^7 + (168
*a^5*b^5*x^(15/2))/5 + (105*a^4*b^6*x^8)/4 + (240*a^3*b^7*x^(17/2))/17 + 5*a^2*b^8*x^9 + (20*a*b^9*x^(19/2))/1
9 + (b^10*x^10)/10

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fricas [A]  time = 1.03, size = 118, normalized size = 0.84 \[ \frac {1}{10} \, b^{10} x^{10} + 5 \, a^{2} b^{8} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 30 \, a^{6} b^{4} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + \frac {1}{5} \, a^{10} x^{5} + \frac {4}{230945} \, {\left (60775 \, a b^{9} x^{9} + 815100 \, a^{3} b^{7} x^{8} + 1939938 \, a^{5} b^{5} x^{7} + 1065900 \, a^{7} b^{3} x^{6} + 104975 \, a^{9} b x^{5}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^10,x, algorithm="fricas")

[Out]

1/10*b^10*x^10 + 5*a^2*b^8*x^9 + 105/4*a^4*b^6*x^8 + 30*a^6*b^4*x^7 + 15/2*a^8*b^2*x^6 + 1/5*a^10*x^5 + 4/2309
45*(60775*a*b^9*x^9 + 815100*a^3*b^7*x^8 + 1939938*a^5*b^5*x^7 + 1065900*a^7*b^3*x^6 + 104975*a^9*b*x^5)*sqrt(
x)

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giac [A]  time = 0.16, size = 112, normalized size = 0.80 \[ \frac {1}{10} \, b^{10} x^{10} + \frac {20}{19} \, a b^{9} x^{\frac {19}{2}} + 5 \, a^{2} b^{8} x^{9} + \frac {240}{17} \, a^{3} b^{7} x^{\frac {17}{2}} + \frac {105}{4} \, a^{4} b^{6} x^{8} + \frac {168}{5} \, a^{5} b^{5} x^{\frac {15}{2}} + 30 \, a^{6} b^{4} x^{7} + \frac {240}{13} \, a^{7} b^{3} x^{\frac {13}{2}} + \frac {15}{2} \, a^{8} b^{2} x^{6} + \frac {20}{11} \, a^{9} b x^{\frac {11}{2}} + \frac {1}{5} \, a^{10} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^10,x, algorithm="giac")

[Out]

1/10*b^10*x^10 + 20/19*a*b^9*x^(19/2) + 5*a^2*b^8*x^9 + 240/17*a^3*b^7*x^(17/2) + 105/4*a^4*b^6*x^8 + 168/5*a^
5*b^5*x^(15/2) + 30*a^6*b^4*x^7 + 240/13*a^7*b^3*x^(13/2) + 15/2*a^8*b^2*x^6 + 20/11*a^9*b*x^(11/2) + 1/5*a^10
*x^5

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maple [A]  time = 0.00, size = 113, normalized size = 0.81 \[ \frac {b^{10} x^{10}}{10}+\frac {20 a \,b^{9} x^{\frac {19}{2}}}{19}+5 a^{2} b^{8} x^{9}+\frac {240 a^{3} b^{7} x^{\frac {17}{2}}}{17}+\frac {105 a^{4} b^{6} x^{8}}{4}+\frac {168 a^{5} b^{5} x^{\frac {15}{2}}}{5}+30 a^{6} b^{4} x^{7}+\frac {240 a^{7} b^{3} x^{\frac {13}{2}}}{13}+\frac {15 a^{8} b^{2} x^{6}}{2}+\frac {20 a^{9} b \,x^{\frac {11}{2}}}{11}+\frac {a^{10} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*x^(1/2))^10,x)

[Out]

1/5*a^10*x^5+20/11*a^9*b*x^(11/2)+15/2*a^8*b^2*x^6+240/13*a^7*b^3*x^(13/2)+30*a^6*b^4*x^7+168/5*a^5*b^5*x^(15/
2)+105/4*a^4*b^6*x^8+240/17*a^3*b^7*x^(17/2)+5*a^2*b^8*x^9+20/19*a*b^9*x^(19/2)+1/10*b^10*x^10

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maxima [A]  time = 0.93, size = 166, normalized size = 1.19 \[ \frac {{\left (b \sqrt {x} + a\right )}^{20}}{10 \, b^{10}} - \frac {18 \, {\left (b \sqrt {x} + a\right )}^{19} a}{19 \, b^{10}} + \frac {4 \, {\left (b \sqrt {x} + a\right )}^{18} a^{2}}{b^{10}} - \frac {168 \, {\left (b \sqrt {x} + a\right )}^{17} a^{3}}{17 \, b^{10}} + \frac {63 \, {\left (b \sqrt {x} + a\right )}^{16} a^{4}}{4 \, b^{10}} - \frac {84 \, {\left (b \sqrt {x} + a\right )}^{15} a^{5}}{5 \, b^{10}} + \frac {12 \, {\left (b \sqrt {x} + a\right )}^{14} a^{6}}{b^{10}} - \frac {72 \, {\left (b \sqrt {x} + a\right )}^{13} a^{7}}{13 \, b^{10}} + \frac {3 \, {\left (b \sqrt {x} + a\right )}^{12} a^{8}}{2 \, b^{10}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{11} a^{9}}{11 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^10,x, algorithm="maxima")

[Out]

1/10*(b*sqrt(x) + a)^20/b^10 - 18/19*(b*sqrt(x) + a)^19*a/b^10 + 4*(b*sqrt(x) + a)^18*a^2/b^10 - 168/17*(b*sqr
t(x) + a)^17*a^3/b^10 + 63/4*(b*sqrt(x) + a)^16*a^4/b^10 - 84/5*(b*sqrt(x) + a)^15*a^5/b^10 + 12*(b*sqrt(x) +
a)^14*a^6/b^10 - 72/13*(b*sqrt(x) + a)^13*a^7/b^10 + 3/2*(b*sqrt(x) + a)^12*a^8/b^10 - 2/11*(b*sqrt(x) + a)^11
*a^9/b^10

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mupad [B]  time = 1.09, size = 112, normalized size = 0.80 \[ \frac {a^{10}\,x^5}{5}+\frac {b^{10}\,x^{10}}{10}+\frac {20\,a^9\,b\,x^{11/2}}{11}+\frac {20\,a\,b^9\,x^{19/2}}{19}+\frac {15\,a^8\,b^2\,x^6}{2}+30\,a^6\,b^4\,x^7+\frac {105\,a^4\,b^6\,x^8}{4}+5\,a^2\,b^8\,x^9+\frac {240\,a^7\,b^3\,x^{13/2}}{13}+\frac {168\,a^5\,b^5\,x^{15/2}}{5}+\frac {240\,a^3\,b^7\,x^{17/2}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^(1/2))^10,x)

[Out]

(a^10*x^5)/5 + (b^10*x^10)/10 + (20*a^9*b*x^(11/2))/11 + (20*a*b^9*x^(19/2))/19 + (15*a^8*b^2*x^6)/2 + 30*a^6*
b^4*x^7 + (105*a^4*b^6*x^8)/4 + 5*a^2*b^8*x^9 + (240*a^7*b^3*x^(13/2))/13 + (168*a^5*b^5*x^(15/2))/5 + (240*a^
3*b^7*x^(17/2))/17

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sympy [A]  time = 7.82, size = 139, normalized size = 0.99 \[ \frac {a^{10} x^{5}}{5} + \frac {20 a^{9} b x^{\frac {11}{2}}}{11} + \frac {15 a^{8} b^{2} x^{6}}{2} + \frac {240 a^{7} b^{3} x^{\frac {13}{2}}}{13} + 30 a^{6} b^{4} x^{7} + \frac {168 a^{5} b^{5} x^{\frac {15}{2}}}{5} + \frac {105 a^{4} b^{6} x^{8}}{4} + \frac {240 a^{3} b^{7} x^{\frac {17}{2}}}{17} + 5 a^{2} b^{8} x^{9} + \frac {20 a b^{9} x^{\frac {19}{2}}}{19} + \frac {b^{10} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*x**(1/2))**10,x)

[Out]

a**10*x**5/5 + 20*a**9*b*x**(11/2)/11 + 15*a**8*b**2*x**6/2 + 240*a**7*b**3*x**(13/2)/13 + 30*a**6*b**4*x**7 +
168*a**5*b**5*x**(15/2)/5 + 105*a**4*b**6*x**8/4 + 240*a**3*b**7*x**(17/2)/17 + 5*a**2*b**8*x**9 + 20*a*b**9*
x**(19/2)/19 + b**10*x**10/10

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