3.2141 \(\int (a+b \sqrt {x})^5 x^4 \, dx\)

Optimal. Leaf size=75 \[ \frac {a^5 x^5}{5}+\frac {10}{11} a^4 b x^{11/2}+\frac {5}{3} a^3 b^2 x^6+\frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2} \]

[Out]

1/5*a^5*x^5+10/11*a^4*b*x^(11/2)+5/3*a^3*b^2*x^6+20/13*a^2*b^3*x^(13/2)+5/7*a*b^4*x^7+2/15*b^5*x^(15/2)

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Rubi [A]  time = 0.05, antiderivative size = 75, 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} \[ \frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{3} a^3 b^2 x^6+\frac {10}{11} a^4 b x^{11/2}+\frac {a^5 x^5}{5}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5*x^4,x]

[Out]

(a^5*x^5)/5 + (10*a^4*b*x^(11/2))/11 + (5*a^3*b^2*x^6)/3 + (20*a^2*b^3*x^(13/2))/13 + (5*a*b^4*x^7)/7 + (2*b^5
*x^(15/2))/15

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 )^5 x^4 \, dx &=2 \operatorname {Subst}\left (\int x^9 (a+b x)^5 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^5 x^9+5 a^4 b x^{10}+10 a^3 b^2 x^{11}+10 a^2 b^3 x^{12}+5 a b^4 x^{13}+b^5 x^{14}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^5 x^5}{5}+\frac {10}{11} a^4 b x^{11/2}+\frac {5}{3} a^3 b^2 x^6+\frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 75, normalized size = 1.00 \[ \frac {a^5 x^5}{5}+\frac {10}{11} a^4 b x^{11/2}+\frac {5}{3} a^3 b^2 x^6+\frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5*x^4,x]

[Out]

(a^5*x^5)/5 + (10*a^4*b*x^(11/2))/11 + (5*a^3*b^2*x^6)/3 + (20*a^2*b^3*x^(13/2))/13 + (5*a*b^4*x^7)/7 + (2*b^5
*x^(15/2))/15

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fricas [A]  time = 0.95, size = 63, normalized size = 0.84 \[ \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {1}{5} \, a^{5} x^{5} + \frac {2}{2145} \, {\left (143 \, b^{5} x^{7} + 1650 \, a^{2} b^{3} x^{6} + 975 \, a^{4} b x^{5}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

5/7*a*b^4*x^7 + 5/3*a^3*b^2*x^6 + 1/5*a^5*x^5 + 2/2145*(143*b^5*x^7 + 1650*a^2*b^3*x^6 + 975*a^4*b*x^5)*sqrt(x
)

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giac [A]  time = 0.15, size = 57, normalized size = 0.76 \[ \frac {2}{15} \, b^{5} x^{\frac {15}{2}} + \frac {5}{7} \, a b^{4} x^{7} + \frac {20}{13} \, a^{2} b^{3} x^{\frac {13}{2}} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {10}{11} \, a^{4} b x^{\frac {11}{2}} + \frac {1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/15*b^5*x^(15/2) + 5/7*a*b^4*x^7 + 20/13*a^2*b^3*x^(13/2) + 5/3*a^3*b^2*x^6 + 10/11*a^4*b*x^(11/2) + 1/5*a^5*
x^5

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maple [A]  time = 0.00, size = 58, normalized size = 0.77 \[ \frac {2 b^{5} x^{\frac {15}{2}}}{15}+\frac {5 a \,b^{4} x^{7}}{7}+\frac {20 a^{2} b^{3} x^{\frac {13}{2}}}{13}+\frac {5 a^{3} b^{2} x^{6}}{3}+\frac {10 a^{4} b \,x^{\frac {11}{2}}}{11}+\frac {a^{5} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*x^(1/2))^5,x)

[Out]

1/5*a^5*x^5+10/11*a^4*b*x^(11/2)+5/3*a^3*b^2*x^6+20/13*a^2*b^3*x^(13/2)+5/7*a*b^4*x^7+2/15*b^5*x^(15/2)

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maxima [B]  time = 0.94, size = 166, normalized size = 2.21 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{15}}{15 \, b^{10}} - \frac {9 \, {\left (b \sqrt {x} + a\right )}^{14} a}{7 \, b^{10}} + \frac {72 \, {\left (b \sqrt {x} + a\right )}^{13} a^{2}}{13 \, b^{10}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{12} a^{3}}{b^{10}} + \frac {252 \, {\left (b \sqrt {x} + a\right )}^{11} a^{4}}{11 \, b^{10}} - \frac {126 \, {\left (b \sqrt {x} + a\right )}^{10} a^{5}}{5 \, b^{10}} + \frac {56 \, {\left (b \sqrt {x} + a\right )}^{9} a^{6}}{3 \, b^{10}} - \frac {9 \, {\left (b \sqrt {x} + a\right )}^{8} a^{7}}{b^{10}} + \frac {18 \, {\left (b \sqrt {x} + a\right )}^{7} a^{8}}{7 \, b^{10}} - \frac {{\left (b \sqrt {x} + a\right )}^{6} a^{9}}{3 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/15*(b*sqrt(x) + a)^15/b^10 - 9/7*(b*sqrt(x) + a)^14*a/b^10 + 72/13*(b*sqrt(x) + a)^13*a^2/b^10 - 14*(b*sqrt(
x) + a)^12*a^3/b^10 + 252/11*(b*sqrt(x) + a)^11*a^4/b^10 - 126/5*(b*sqrt(x) + a)^10*a^5/b^10 + 56/3*(b*sqrt(x)
+ a)^9*a^6/b^10 - 9*(b*sqrt(x) + a)^8*a^7/b^10 + 18/7*(b*sqrt(x) + a)^7*a^8/b^10 - 1/3*(b*sqrt(x) + a)^6*a^9/
b^10

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mupad [B]  time = 0.03, size = 57, normalized size = 0.76 \[ \frac {a^5\,x^5}{5}+\frac {2\,b^5\,x^{15/2}}{15}+\frac {5\,a\,b^4\,x^7}{7}+\frac {10\,a^4\,b\,x^{11/2}}{11}+\frac {5\,a^3\,b^2\,x^6}{3}+\frac {20\,a^2\,b^3\,x^{13/2}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^(1/2))^5,x)

[Out]

(a^5*x^5)/5 + (2*b^5*x^(15/2))/15 + (5*a*b^4*x^7)/7 + (10*a^4*b*x^(11/2))/11 + (5*a^3*b^2*x^6)/3 + (20*a^2*b^3
*x^(13/2))/13

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sympy [A]  time = 2.73, size = 73, normalized size = 0.97 \[ \frac {a^{5} x^{5}}{5} + \frac {10 a^{4} b x^{\frac {11}{2}}}{11} + \frac {5 a^{3} b^{2} x^{6}}{3} + \frac {20 a^{2} b^{3} x^{\frac {13}{2}}}{13} + \frac {5 a b^{4} x^{7}}{7} + \frac {2 b^{5} x^{\frac {15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**5/5 + 10*a**4*b*x**(11/2)/11 + 5*a**3*b**2*x**6/3 + 20*a**2*b**3*x**(13/2)/13 + 5*a*b**4*x**7/7 + 2*b*
*5*x**(15/2)/15

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