3.2142 \(\int (a+b \sqrt {x})^5 x^3 \, dx\)

Optimal. Leaf size=73 \[ \frac {a^5 x^4}{4}+\frac {10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac {20}{11} a^2 b^3 x^{11/2}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2} \]

[Out]

1/4*a^5*x^4+10/9*a^4*b*x^(9/2)+2*a^3*b^2*x^5+20/11*a^2*b^3*x^(11/2)+5/6*a*b^4*x^6+2/13*b^5*x^(13/2)

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Rubi [A]  time = 0.04, antiderivative size = 73, 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} \[ \frac {20}{11} a^2 b^3 x^{11/2}+2 a^3 b^2 x^5+\frac {10}{9} a^4 b x^{9/2}+\frac {a^5 x^4}{4}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5*x^3,x]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 + (5*a*b^4*x^6)/6 + (2*b^5*x^(13
/2))/13

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^3 \, dx &=2 \operatorname {Subst}\left (\int x^7 (a+b x)^5 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^5 x^7+5 a^4 b x^8+10 a^3 b^2 x^9+10 a^2 b^3 x^{10}+5 a b^4 x^{11}+b^5 x^{12}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^5 x^4}{4}+\frac {10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac {20}{11} a^2 b^3 x^{11/2}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 73, normalized size = 1.00 \[ \frac {a^5 x^4}{4}+\frac {10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac {20}{11} a^2 b^3 x^{11/2}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5*x^3,x]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 + (5*a*b^4*x^6)/6 + (2*b^5*x^(13
/2))/13

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fricas [A]  time = 1.00, size = 63, normalized size = 0.86 \[ \frac {5}{6} \, a b^{4} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac {1}{4} \, a^{5} x^{4} + \frac {2}{1287} \, {\left (99 \, b^{5} x^{6} + 1170 \, a^{2} b^{3} x^{5} + 715 \, a^{4} b x^{4}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

5/6*a*b^4*x^6 + 2*a^3*b^2*x^5 + 1/4*a^5*x^4 + 2/1287*(99*b^5*x^6 + 1170*a^2*b^3*x^5 + 715*a^4*b*x^4)*sqrt(x)

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giac [A]  time = 0.16, size = 57, normalized size = 0.78 \[ \frac {2}{13} \, b^{5} x^{\frac {13}{2}} + \frac {5}{6} \, a b^{4} x^{6} + \frac {20}{11} \, a^{2} b^{3} x^{\frac {11}{2}} + 2 \, a^{3} b^{2} x^{5} + \frac {10}{9} \, a^{4} b x^{\frac {9}{2}} + \frac {1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/13*b^5*x^(13/2) + 5/6*a*b^4*x^6 + 20/11*a^2*b^3*x^(11/2) + 2*a^3*b^2*x^5 + 10/9*a^4*b*x^(9/2) + 1/4*a^5*x^4

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maple [A]  time = 0.00, size = 58, normalized size = 0.79 \[ \frac {2 b^{5} x^{\frac {13}{2}}}{13}+\frac {5 a \,b^{4} x^{6}}{6}+\frac {20 a^{2} b^{3} x^{\frac {11}{2}}}{11}+2 a^{3} b^{2} x^{5}+\frac {10 a^{4} b \,x^{\frac {9}{2}}}{9}+\frac {a^{5} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*x^(1/2))^5,x)

[Out]

1/4*a^5*x^4+10/9*a^4*b*x^(9/2)+2*a^3*b^2*x^5+20/11*a^2*b^3*x^(11/2)+5/6*a*b^4*x^6+2/13*b^5*x^(13/2)

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maxima [B]  time = 0.90, size = 132, normalized size = 1.81 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{13}}{13 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{12} a}{6 \, b^{8}} + \frac {42 \, {\left (b \sqrt {x} + a\right )}^{11} a^{2}}{11 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{10} a^{3}}{b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )}^{9} a^{4}}{9 \, b^{8}} - \frac {21 \, {\left (b \sqrt {x} + a\right )}^{8} a^{5}}{4 \, b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{7} a^{6}}{b^{8}} - \frac {{\left (b \sqrt {x} + a\right )}^{6} a^{7}}{3 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/13*(b*sqrt(x) + a)^13/b^8 - 7/6*(b*sqrt(x) + a)^12*a/b^8 + 42/11*(b*sqrt(x) + a)^11*a^2/b^8 - 7*(b*sqrt(x) +
a)^10*a^3/b^8 + 70/9*(b*sqrt(x) + a)^9*a^4/b^8 - 21/4*(b*sqrt(x) + a)^8*a^5/b^8 + 2*(b*sqrt(x) + a)^7*a^6/b^8
- 1/3*(b*sqrt(x) + a)^6*a^7/b^8

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mupad [B]  time = 0.03, size = 57, normalized size = 0.78 \[ \frac {a^5\,x^4}{4}+\frac {2\,b^5\,x^{13/2}}{13}+\frac {5\,a\,b^4\,x^6}{6}+\frac {10\,a^4\,b\,x^{9/2}}{9}+2\,a^3\,b^2\,x^5+\frac {20\,a^2\,b^3\,x^{11/2}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^(1/2))^5,x)

[Out]

(a^5*x^4)/4 + (2*b^5*x^(13/2))/13 + (5*a*b^4*x^6)/6 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11
/2))/11

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sympy [A]  time = 2.56, size = 71, normalized size = 0.97 \[ \frac {a^{5} x^{4}}{4} + \frac {10 a^{4} b x^{\frac {9}{2}}}{9} + 2 a^{3} b^{2} x^{5} + \frac {20 a^{2} b^{3} x^{\frac {11}{2}}}{11} + \frac {5 a b^{4} x^{6}}{6} + \frac {2 b^{5} x^{\frac {13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**4/4 + 10*a**4*b*x**(9/2)/9 + 2*a**3*b**2*x**5 + 20*a**2*b**3*x**(11/2)/11 + 5*a*b**4*x**6/6 + 2*b**5*x
**(13/2)/13

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