3.2183 \(\int \frac {(a+b \sqrt {x})^{15}}{x^{10}} \, dx\)

Optimal. Leaf size=70 \[ -\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{1224 a^3 x^8}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{153 a^2 x^{17/2}}-\frac {\left (a+b \sqrt {x}\right )^{16}}{9 a x^9} \]

[Out]

-1/9*(a+b*x^(1/2))^16/a/x^9+2/153*b*(a+b*x^(1/2))^16/a^2/x^(17/2)-1/1224*b^2*(a+b*x^(1/2))^16/a^3/x^8

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Rubi [A]  time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ -\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{1224 a^3 x^8}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{153 a^2 x^{17/2}}-\frac {\left (a+b \sqrt {x}\right )^{16}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^10,x]

[Out]

-(a + b*Sqrt[x])^16/(9*a*x^9) + (2*b*(a + b*Sqrt[x])^16)/(153*a^2*x^(17/2)) - (b^2*(a + b*Sqrt[x])^16)/(1224*a
^3*x^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{9 a x^9}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt {x}\right )}{9 a}\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{9 a x^9}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{153 a^2 x^{17/2}}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt {x}\right )}{153 a^2}\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{9 a x^9}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{153 a^2 x^{17/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{1224 a^3 x^8}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.59 \[ -\frac {\left (a+b \sqrt {x}\right )^{16} \left (136 a^2-16 a b \sqrt {x}+b^2 x\right )}{1224 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^10,x]

[Out]

-1/1224*((a + b*Sqrt[x])^16*(136*a^2 - 16*a*b*Sqrt[x] + b^2*x))/(a^3*x^9)

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fricas [B]  time = 0.98, size = 168, normalized size = 2.40 \[ -\frac {9180 \, a b^{14} x^{7} + 185640 \, a^{3} b^{12} x^{6} + 918918 \, a^{5} b^{10} x^{5} + 1575288 \, a^{7} b^{8} x^{4} + 1021020 \, a^{9} b^{6} x^{3} + 238680 \, a^{11} b^{4} x^{2} + 16065 \, a^{13} b^{2} x + 136 \, a^{15} + 16 \, {\left (51 \, b^{15} x^{7} + 3213 \, a^{2} b^{13} x^{6} + 29835 \, a^{4} b^{11} x^{5} + 85085 \, a^{6} b^{9} x^{4} + 89505 \, a^{8} b^{7} x^{3} + 35343 \, a^{10} b^{5} x^{2} + 4641 \, a^{12} b^{3} x + 135 \, a^{14} b\right )} \sqrt {x}}{1224 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^10,x, algorithm="fricas")

[Out]

-1/1224*(9180*a*b^14*x^7 + 185640*a^3*b^12*x^6 + 918918*a^5*b^10*x^5 + 1575288*a^7*b^8*x^4 + 1021020*a^9*b^6*x
^3 + 238680*a^11*b^4*x^2 + 16065*a^13*b^2*x + 136*a^15 + 16*(51*b^15*x^7 + 3213*a^2*b^13*x^6 + 29835*a^4*b^11*
x^5 + 85085*a^6*b^9*x^4 + 89505*a^8*b^7*x^3 + 35343*a^10*b^5*x^2 + 4641*a^12*b^3*x + 135*a^14*b)*sqrt(x))/x^9

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giac [B]  time = 0.17, size = 167, normalized size = 2.39 \[ -\frac {816 \, b^{15} x^{\frac {15}{2}} + 9180 \, a b^{14} x^{7} + 51408 \, a^{2} b^{13} x^{\frac {13}{2}} + 185640 \, a^{3} b^{12} x^{6} + 477360 \, a^{4} b^{11} x^{\frac {11}{2}} + 918918 \, a^{5} b^{10} x^{5} + 1361360 \, a^{6} b^{9} x^{\frac {9}{2}} + 1575288 \, a^{7} b^{8} x^{4} + 1432080 \, a^{8} b^{7} x^{\frac {7}{2}} + 1021020 \, a^{9} b^{6} x^{3} + 565488 \, a^{10} b^{5} x^{\frac {5}{2}} + 238680 \, a^{11} b^{4} x^{2} + 74256 \, a^{12} b^{3} x^{\frac {3}{2}} + 16065 \, a^{13} b^{2} x + 2160 \, a^{14} b \sqrt {x} + 136 \, a^{15}}{1224 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^10,x, algorithm="giac")

[Out]

-1/1224*(816*b^15*x^(15/2) + 9180*a*b^14*x^7 + 51408*a^2*b^13*x^(13/2) + 185640*a^3*b^12*x^6 + 477360*a^4*b^11
*x^(11/2) + 918918*a^5*b^10*x^5 + 1361360*a^6*b^9*x^(9/2) + 1575288*a^7*b^8*x^4 + 1432080*a^8*b^7*x^(7/2) + 10
21020*a^9*b^6*x^3 + 565488*a^10*b^5*x^(5/2) + 238680*a^11*b^4*x^2 + 74256*a^12*b^3*x^(3/2) + 16065*a^13*b^2*x
+ 2160*a^14*b*sqrt(x) + 136*a^15)/x^9

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maple [B]  time = 0.00, size = 168, normalized size = 2.40 \[ -\frac {2 b^{15}}{3 x^{\frac {3}{2}}}-\frac {15 a \,b^{14}}{2 x^{2}}-\frac {42 a^{2} b^{13}}{x^{\frac {5}{2}}}-\frac {455 a^{3} b^{12}}{3 x^{3}}-\frac {390 a^{4} b^{11}}{x^{\frac {7}{2}}}-\frac {3003 a^{5} b^{10}}{4 x^{4}}-\frac {10010 a^{6} b^{9}}{9 x^{\frac {9}{2}}}-\frac {1287 a^{7} b^{8}}{x^{5}}-\frac {1170 a^{8} b^{7}}{x^{\frac {11}{2}}}-\frac {5005 a^{9} b^{6}}{6 x^{6}}-\frac {462 a^{10} b^{5}}{x^{\frac {13}{2}}}-\frac {195 a^{11} b^{4}}{x^{7}}-\frac {182 a^{12} b^{3}}{3 x^{\frac {15}{2}}}-\frac {105 a^{13} b^{2}}{8 x^{8}}-\frac {30 a^{14} b}{17 x^{\frac {17}{2}}}-\frac {a^{15}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^10,x)

[Out]

-2/3*b^15/x^(3/2)-15/2*a*b^14/x^2-42*a^2*b^13/x^(5/2)-455/3*a^3*b^12/x^3-390*a^4*b^11/x^(7/2)-3003/4*a^5*b^10/
x^4-10010/9*a^6*b^9/x^(9/2)-1287*a^7*b^8/x^5-1170*a^8*b^7/x^(11/2)-5005/6*a^9*b^6/x^6-462*a^10*b^5/x^(13/2)-19
5*a^11*b^4/x^7-182/3*a^12*b^3/x^(15/2)-105/8*a^13*b^2/x^8-30/17*a^14*b/x^(17/2)-1/9*a^15/x^9

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maxima [B]  time = 0.91, size = 167, normalized size = 2.39 \[ -\frac {816 \, b^{15} x^{\frac {15}{2}} + 9180 \, a b^{14} x^{7} + 51408 \, a^{2} b^{13} x^{\frac {13}{2}} + 185640 \, a^{3} b^{12} x^{6} + 477360 \, a^{4} b^{11} x^{\frac {11}{2}} + 918918 \, a^{5} b^{10} x^{5} + 1361360 \, a^{6} b^{9} x^{\frac {9}{2}} + 1575288 \, a^{7} b^{8} x^{4} + 1432080 \, a^{8} b^{7} x^{\frac {7}{2}} + 1021020 \, a^{9} b^{6} x^{3} + 565488 \, a^{10} b^{5} x^{\frac {5}{2}} + 238680 \, a^{11} b^{4} x^{2} + 74256 \, a^{12} b^{3} x^{\frac {3}{2}} + 16065 \, a^{13} b^{2} x + 2160 \, a^{14} b \sqrt {x} + 136 \, a^{15}}{1224 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^10,x, algorithm="maxima")

[Out]

-1/1224*(816*b^15*x^(15/2) + 9180*a*b^14*x^7 + 51408*a^2*b^13*x^(13/2) + 185640*a^3*b^12*x^6 + 477360*a^4*b^11
*x^(11/2) + 918918*a^5*b^10*x^5 + 1361360*a^6*b^9*x^(9/2) + 1575288*a^7*b^8*x^4 + 1432080*a^8*b^7*x^(7/2) + 10
21020*a^9*b^6*x^3 + 565488*a^10*b^5*x^(5/2) + 238680*a^11*b^4*x^2 + 74256*a^12*b^3*x^(3/2) + 16065*a^13*b^2*x
+ 2160*a^14*b*sqrt(x) + 136*a^15)/x^9

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mupad [B]  time = 1.24, size = 167, normalized size = 2.39 \[ -\frac {\frac {a^{15}}{9}+\frac {2\,b^{15}\,x^{15/2}}{3}+\frac {105\,a^{13}\,b^2\,x}{8}+\frac {30\,a^{14}\,b\,\sqrt {x}}{17}+\frac {15\,a\,b^{14}\,x^7}{2}+195\,a^{11}\,b^4\,x^2+\frac {5005\,a^9\,b^6\,x^3}{6}+1287\,a^7\,b^8\,x^4+\frac {3003\,a^5\,b^{10}\,x^5}{4}+\frac {182\,a^{12}\,b^3\,x^{3/2}}{3}+\frac {455\,a^3\,b^{12}\,x^6}{3}+462\,a^{10}\,b^5\,x^{5/2}+1170\,a^8\,b^7\,x^{7/2}+\frac {10010\,a^6\,b^9\,x^{9/2}}{9}+390\,a^4\,b^{11}\,x^{11/2}+42\,a^2\,b^{13}\,x^{13/2}}{x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^15/x^10,x)

[Out]

-(a^15/9 + (2*b^15*x^(15/2))/3 + (105*a^13*b^2*x)/8 + (30*a^14*b*x^(1/2))/17 + (15*a*b^14*x^7)/2 + 195*a^11*b^
4*x^2 + (5005*a^9*b^6*x^3)/6 + 1287*a^7*b^8*x^4 + (3003*a^5*b^10*x^5)/4 + (182*a^12*b^3*x^(3/2))/3 + (455*a^3*
b^12*x^6)/3 + 462*a^10*b^5*x^(5/2) + 1170*a^8*b^7*x^(7/2) + (10010*a^6*b^9*x^(9/2))/9 + 390*a^4*b^11*x^(11/2)
+ 42*a^2*b^13*x^(13/2))/x^9

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sympy [B]  time = 10.07, size = 209, normalized size = 2.99 \[ - \frac {a^{15}}{9 x^{9}} - \frac {30 a^{14} b}{17 x^{\frac {17}{2}}} - \frac {105 a^{13} b^{2}}{8 x^{8}} - \frac {182 a^{12} b^{3}}{3 x^{\frac {15}{2}}} - \frac {195 a^{11} b^{4}}{x^{7}} - \frac {462 a^{10} b^{5}}{x^{\frac {13}{2}}} - \frac {5005 a^{9} b^{6}}{6 x^{6}} - \frac {1170 a^{8} b^{7}}{x^{\frac {11}{2}}} - \frac {1287 a^{7} b^{8}}{x^{5}} - \frac {10010 a^{6} b^{9}}{9 x^{\frac {9}{2}}} - \frac {3003 a^{5} b^{10}}{4 x^{4}} - \frac {390 a^{4} b^{11}}{x^{\frac {7}{2}}} - \frac {455 a^{3} b^{12}}{3 x^{3}} - \frac {42 a^{2} b^{13}}{x^{\frac {5}{2}}} - \frac {15 a b^{14}}{2 x^{2}} - \frac {2 b^{15}}{3 x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**10,x)

[Out]

-a**15/(9*x**9) - 30*a**14*b/(17*x**(17/2)) - 105*a**13*b**2/(8*x**8) - 182*a**12*b**3/(3*x**(15/2)) - 195*a**
11*b**4/x**7 - 462*a**10*b**5/x**(13/2) - 5005*a**9*b**6/(6*x**6) - 1170*a**8*b**7/x**(11/2) - 1287*a**7*b**8/
x**5 - 10010*a**6*b**9/(9*x**(9/2)) - 3003*a**5*b**10/(4*x**4) - 390*a**4*b**11/x**(7/2) - 455*a**3*b**12/(3*x
**3) - 42*a**2*b**13/x**(5/2) - 15*a*b**14/(2*x**2) - 2*b**15/(3*x**(3/2))

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