3.37 \(\int \frac {(a x+b x^2)^{5/2}}{x^{11}} \, dx\)

Optimal. Leaf size=126 \[ -\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac {128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac {16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}} \]

[Out]

-2/15*(b*x^2+a*x)^(7/2)/a/x^11+16/195*b*(b*x^2+a*x)^(7/2)/a^2/x^10-32/715*b^2*(b*x^2+a*x)^(7/2)/a^3/x^9+128/64
35*b^3*(b*x^2+a*x)^(7/2)/a^4/x^8-256/45045*b^4*(b*x^2+a*x)^(7/2)/a^5/x^7

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Rubi [A]  time = 0.06, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {658, 650} \[ -\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac {128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac {16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^2)^(5/2)/x^11,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(15*a*x^11) + (16*b*(a*x + b*x^2)^(7/2))/(195*a^2*x^10) - (32*b^2*(a*x + b*x^2)^(7/2)
)/(715*a^3*x^9) + (128*b^3*(a*x + b*x^2)^(7/2))/(6435*a^4*x^8) - (256*b^4*(a*x + b*x^2)^(7/2))/(45045*a^5*x^7)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rubi steps

\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{11}} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}-\frac {(8 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx}{15 a}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac {16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}+\frac {\left (16 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{65 a^2}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac {16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}-\frac {\left (64 b^3\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{715 a^3}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac {16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac {128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}+\frac {\left (128 b^4\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{6435 a^4}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac {16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac {128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 69, normalized size = 0.55 \[ -\frac {2 (a+b x)^3 \sqrt {x (a+b x)} \left (3003 a^4-1848 a^3 b x+1008 a^2 b^2 x^2-448 a b^3 x^3+128 b^4 x^4\right )}{45045 a^5 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^11,x]

[Out]

(-2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(3003*a^4 - 1848*a^3*b*x + 1008*a^2*b^2*x^2 - 448*a*b^3*x^3 + 128*b^4*x^4))/
(45045*a^5*x^8)

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fricas [A]  time = 0.99, size = 93, normalized size = 0.74 \[ -\frac {2 \, {\left (128 \, b^{7} x^{7} - 64 \, a b^{6} x^{6} + 48 \, a^{2} b^{5} x^{5} - 40 \, a^{3} b^{4} x^{4} + 35 \, a^{4} b^{3} x^{3} + 4473 \, a^{5} b^{2} x^{2} + 7161 \, a^{6} b x + 3003 \, a^{7}\right )} \sqrt {b x^{2} + a x}}{45045 \, a^{5} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^11,x, algorithm="fricas")

[Out]

-2/45045*(128*b^7*x^7 - 64*a*b^6*x^6 + 48*a^2*b^5*x^5 - 40*a^3*b^4*x^4 + 35*a^4*b^3*x^3 + 4473*a^5*b^2*x^2 + 7
161*a^6*b*x + 3003*a^7)*sqrt(b*x^2 + a*x)/(a^5*x^8)

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giac [B]  time = 0.22, size = 310, normalized size = 2.46 \[ \frac {2 \, {\left (144144 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{10} b^{5} + 960960 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} a b^{\frac {9}{2}} + 2934360 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a^{2} b^{4} + 5360355 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{3} b^{\frac {7}{2}} + 6451445 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{4} b^{3} + 5324319 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{5} b^{\frac {5}{2}} + 3042585 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{6} b^{2} + 1186185 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{7} b^{\frac {3}{2}} + 301455 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{8} b + 45045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{9} \sqrt {b} + 3003 \, a^{10}\right )}}{45045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^11,x, algorithm="giac")

[Out]

2/45045*(144144*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*b^5 + 960960*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a*b^(9/2) +
2934360*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^2*b^4 + 5360355*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^3*b^(7/2) + 64
51445*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^4*b^3 + 5324319*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^5*b^(5/2) + 3042
585*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^6*b^2 + 1186185*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^7*b^(3/2) + 301455
*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^8*b + 45045*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^9*sqrt(b) + 3003*a^10)/(sqr
t(b)*x - sqrt(b*x^2 + a*x))^15

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maple [A]  time = 0.05, size = 66, normalized size = 0.52 \[ -\frac {2 \left (b x +a \right ) \left (128 b^{4} x^{4}-448 a \,b^{3} x^{3}+1008 b^{2} x^{2} a^{2}-1848 b x \,a^{3}+3003 a^{4}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{45045 a^{5} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a*x)^(5/2)/x^11,x)

[Out]

-2/45045*(b*x+a)*(128*b^4*x^4-448*a*b^3*x^3+1008*a^2*b^2*x^2-1848*a^3*b*x+3003*a^4)*(b*x^2+a*x)^(5/2)/x^10/a^5

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maxima [A]  time = 1.45, size = 200, normalized size = 1.59 \[ -\frac {256 \, \sqrt {b x^{2} + a x} b^{7}}{45045 \, a^{5} x} + \frac {128 \, \sqrt {b x^{2} + a x} b^{6}}{45045 \, a^{4} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} b^{5}}{15015 \, a^{3} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} b^{4}}{9009 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{1287 \, a x^{5}} + \frac {\sqrt {b x^{2} + a x} b^{2}}{715 \, x^{6}} - \frac {\sqrt {b x^{2} + a x} a b}{780 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} a^{2}}{60 \, x^{8}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{12 \, x^{9}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{5 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^11,x, algorithm="maxima")

[Out]

-256/45045*sqrt(b*x^2 + a*x)*b^7/(a^5*x) + 128/45045*sqrt(b*x^2 + a*x)*b^6/(a^4*x^2) - 32/15015*sqrt(b*x^2 + a
*x)*b^5/(a^3*x^3) + 16/9009*sqrt(b*x^2 + a*x)*b^4/(a^2*x^4) - 2/1287*sqrt(b*x^2 + a*x)*b^3/(a*x^5) + 1/715*sqr
t(b*x^2 + a*x)*b^2/x^6 - 1/780*sqrt(b*x^2 + a*x)*a*b/x^7 - 1/60*sqrt(b*x^2 + a*x)*a^2/x^8 + 1/12*(b*x^2 + a*x)
^(3/2)*a/x^9 - 1/5*(b*x^2 + a*x)^(5/2)/x^10

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mupad [B]  time = 1.86, size = 167, normalized size = 1.33 \[ \frac {16\,b^4\,\sqrt {b\,x^2+a\,x}}{9009\,a^2\,x^4}-\frac {142\,b^2\,\sqrt {b\,x^2+a\,x}}{715\,x^6}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{1287\,a\,x^5}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{15\,x^8}-\frac {32\,b^5\,\sqrt {b\,x^2+a\,x}}{15015\,a^3\,x^3}+\frac {128\,b^6\,\sqrt {b\,x^2+a\,x}}{45045\,a^4\,x^2}-\frac {256\,b^7\,\sqrt {b\,x^2+a\,x}}{45045\,a^5\,x}-\frac {62\,a\,b\,\sqrt {b\,x^2+a\,x}}{195\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^2)^(5/2)/x^11,x)

[Out]

(16*b^4*(a*x + b*x^2)^(1/2))/(9009*a^2*x^4) - (142*b^2*(a*x + b*x^2)^(1/2))/(715*x^6) - (2*b^3*(a*x + b*x^2)^(
1/2))/(1287*a*x^5) - (2*a^2*(a*x + b*x^2)^(1/2))/(15*x^8) - (32*b^5*(a*x + b*x^2)^(1/2))/(15015*a^3*x^3) + (12
8*b^6*(a*x + b*x^2)^(1/2))/(45045*a^4*x^2) - (256*b^7*(a*x + b*x^2)^(1/2))/(45045*a^5*x) - (62*a*b*(a*x + b*x^
2)^(1/2))/(195*x^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**11,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**11, x)

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