3.1.36 \(\int \frac {(a x+b x^2)^{5/2}}{x^{10}} \, dx\) [36]

Optimal. Leaf size=100 \[ -\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7} \]

[Out]

-2/13*(b*x^2+a*x)^(7/2)/a/x^10+12/143*b*(b*x^2+a*x)^(7/2)/a^2/x^9-16/429*b^2*(b*x^2+a*x)^(7/2)/a^3/x^8+32/3003
*b^3*(b*x^2+a*x)^(7/2)/a^4/x^7

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Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \begin {gather*} \frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^2)^(5/2)/x^10,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(13*a*x^10) + (12*b*(a*x + b*x^2)^(7/2))/(143*a^2*x^9) - (16*b^2*(a*x + b*x^2)^(7/2))
/(429*a^3*x^8) + (32*b^3*(a*x + b*x^2)^(7/2))/(3003*a^4*x^7)

Rule 664

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 672

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^{10}} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}-\frac {(6 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{13 a}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}+\frac {\left (24 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{143 a^2}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}-\frac {\left (16 b^3\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{429 a^3}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 51, normalized size = 0.51 \begin {gather*} -\frac {2 (x (a+b x))^{7/2} \left (231 a^3-126 a^2 b x+56 a b^2 x^2-16 b^3 x^3\right )}{3003 a^4 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^10,x]

[Out]

(-2*(x*(a + b*x))^(7/2)*(231*a^3 - 126*a^2*b*x + 56*a*b^2*x^2 - 16*b^3*x^3))/(3003*a^4*x^10)

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Maple [A]
time = 0.42, size = 93, normalized size = 0.93

method	result	size
gosper	\(-\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+56 a \,b^{2} x^{2}-126 a^{2} b x +231 a^{3}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{3003 x^{9} a^{4}}\)	\(55\)
trager	\(-\frac {2 \left (-16 b^{6} x^{6}+8 a \,b^{5} x^{5}-6 a^{2} b^{4} x^{4}+5 a^{3} b^{3} x^{3}+371 a^{4} b^{2} x^{2}+567 a^{5} b x +231 a^{6}\right ) \sqrt {b \,x^{2}+a x}}{3003 a^{4} x^{7}}\)	\(83\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-16 b^{6} x^{6}+8 a \,b^{5} x^{5}-6 a^{2} b^{4} x^{4}+5 a^{3} b^{3} x^{3}+371 a^{4} b^{2} x^{2}+567 a^{5} b x +231 a^{6}\right )}{3003 x^{6} \sqrt {x \left (b x +a \right )}\, a^{4}}\)	\(86\)
default	\(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{13 a \,x^{10}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{11 a \,x^{9}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{9 a \,x^{8}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a*x)^(5/2)/x^10,x,method=_RETURNVERBOSE)

[Out]

-2/13*(b*x^2+a*x)^(7/2)/a/x^10-6/13*b/a*(-2/11*(b*x^2+a*x)^(7/2)/a/x^9-4/11*b/a*(-2/9*(b*x^2+a*x)^(7/2)/a/x^8+
4/63*b*(b*x^2+a*x)^(7/2)/a^2/x^7))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (84) = 168\).
time = 0.29, size = 178, normalized size = 1.78 \begin {gather*} \frac {32 \, \sqrt {b x^{2} + a x} b^{6}}{3003 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{5}}{3003 \, a^{3} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{4}}{1001 \, a^{2} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} b^{3}}{3003 \, a x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} b^{2}}{1716 \, x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} a b}{1144 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} a^{2}}{104 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{8 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{4 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^10,x, algorithm="maxima")

[Out]

32/3003*sqrt(b*x^2 + a*x)*b^6/(a^4*x) - 16/3003*sqrt(b*x^2 + a*x)*b^5/(a^3*x^2) + 4/1001*sqrt(b*x^2 + a*x)*b^4
/(a^2*x^3) - 10/3003*sqrt(b*x^2 + a*x)*b^3/(a*x^4) + 5/1716*sqrt(b*x^2 + a*x)*b^2/x^5 - 3/1144*sqrt(b*x^2 + a*
x)*a*b/x^6 - 3/104*sqrt(b*x^2 + a*x)*a^2/x^7 + 1/8*(b*x^2 + a*x)^(3/2)*a/x^8 - 1/4*(b*x^2 + a*x)^(5/2)/x^9

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Fricas [A]
time = 1.54, size = 82, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt {b x^{2} + a x}}{3003 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^10,x, algorithm="fricas")

[Out]

2/3003*(16*b^6*x^6 - 8*a*b^5*x^5 + 6*a^2*b^4*x^4 - 5*a^3*b^3*x^3 - 371*a^4*b^2*x^2 - 567*a^5*b*x - 231*a^6)*sq
rt(b*x^2 + a*x)/(a^4*x^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{10}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**10,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**10, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (84) = 168\).
time = 0.91, size = 281, normalized size = 2.81 \begin {gather*} \frac {2 \, {\left (6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} b^{\frac {9}{2}} + 36036 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{2} b^{\frac {7}{2}} + 161733 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{4} b^{\frac {5}{2}} + 121121 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{6} b^{\frac {3}{2}} + 17199 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{8} \sqrt {b} + 231 \, a^{9}\right )}}{3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^10,x, algorithm="giac")

[Out]

2/3003*(6006*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*b^(9/2) + 36036*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a*b^4 + 99099
*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^2*b^(7/2) + 161733*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*b^3 + 171171*(sq
rt(b)*x - sqrt(b*x^2 + a*x))^5*a^4*b^(5/2) + 121121*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^5*b^2 + 57057*(sqrt(b)
*x - sqrt(b*x^2 + a*x))^3*a^6*b^(3/2) + 17199*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^7*b + 3003*(sqrt(b)*x - sqrt
(b*x^2 + a*x))*a^8*sqrt(b) + 231*a^9)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^13

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Mupad [B]
time = 1.59, size = 145, normalized size = 1.45 \begin {gather*} \frac {4\,b^4\,\sqrt {b\,x^2+a\,x}}{1001\,a^2\,x^3}-\frac {106\,b^2\,\sqrt {b\,x^2+a\,x}}{429\,x^5}-\frac {10\,b^3\,\sqrt {b\,x^2+a\,x}}{3003\,a\,x^4}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{13\,x^7}-\frac {16\,b^5\,\sqrt {b\,x^2+a\,x}}{3003\,a^3\,x^2}+\frac {32\,b^6\,\sqrt {b\,x^2+a\,x}}{3003\,a^4\,x}-\frac {54\,a\,b\,\sqrt {b\,x^2+a\,x}}{143\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^2)^(5/2)/x^10,x)

[Out]

(4*b^4*(a*x + b*x^2)^(1/2))/(1001*a^2*x^3) - (106*b^2*(a*x + b*x^2)^(1/2))/(429*x^5) - (10*b^3*(a*x + b*x^2)^(
1/2))/(3003*a*x^4) - (2*a^2*(a*x + b*x^2)^(1/2))/(13*x^7) - (16*b^5*(a*x + b*x^2)^(1/2))/(3003*a^3*x^2) + (32*
b^6*(a*x + b*x^2)^(1/2))/(3003*a^4*x) - (54*a*b*(a*x + b*x^2)^(1/2))/(143*x^6)

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