3.1.38 \(\int \frac {(a x+b x^2)^{5/2}}{x^{12}} \, dx\) [38]

Optimal. Leaf size=152 \[ -\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac {512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7} \]

[Out]

-2/17*(b*x^2+a*x)^(7/2)/a/x^12+4/51*b*(b*x^2+a*x)^(7/2)/a^2/x^11-32/663*b^2*(b*x^2+a*x)^(7/2)/a^3/x^10+64/2431
*b^3*(b*x^2+a*x)^(7/2)/a^4/x^9-256/21879*b^4*(b*x^2+a*x)^(7/2)/a^5/x^8+512/153153*b^5*(b*x^2+a*x)^(7/2)/a^6/x^
7

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Rubi [A]
time = 0.05, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^2)^(5/2)/x^12,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(17*a*x^12) + (4*b*(a*x + b*x^2)^(7/2))/(51*a^2*x^11) - (32*b^2*(a*x + b*x^2)^(7/2))/
(663*a^3*x^10) + (64*b^3*(a*x + b*x^2)^(7/2))/(2431*a^4*x^9) - (256*b^4*(a*x + b*x^2)^(7/2))/(21879*a^5*x^8) +
(512*b^5*(a*x + b*x^2)^(7/2))/(153153*a^6*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^{12}} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}-\frac {(10 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{11}} \, dx}{17 a}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}+\frac {\left (16 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx}{51 a^2}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}-\frac {\left (32 b^3\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{221 a^3}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}+\frac {\left (128 b^4\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{2431 a^4}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}-\frac {\left (256 b^5\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{21879 a^5}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac {512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 80, normalized size = 0.53 \begin {gather*} \frac {2 (a+b x)^3 \sqrt {x (a+b x)} \left (-9009 a^5+6006 a^4 b x-3696 a^3 b^2 x^2+2016 a^2 b^3 x^3-896 a b^4 x^4+256 b^5 x^5\right )}{153153 a^6 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^12,x]

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-9009*a^5 + 6006*a^4*b*x - 3696*a^3*b^2*x^2 + 2016*a^2*b^3*x^3 - 896*a*b^4*x
^4 + 256*b^5*x^5))/(153153*a^6*x^9)

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Maple [A]
time = 0.40, size = 145, normalized size = 0.95

method	result	size
gosper	\(-\frac {2 \left (b x +a \right ) \left (-256 b^{5} x^{5}+896 a \,b^{4} x^{4}-2016 a^{2} b^{3} x^{3}+3696 a^{3} x^{2} b^{2}-6006 a^{4} b x +9009 a^{5}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{153153 x^{11} a^{6}}\)	\(77\)
trager	\(-\frac {2 \left (-256 b^{8} x^{8}+128 a \,b^{7} x^{7}-96 a^{2} b^{6} x^{6}+80 a^{3} b^{5} x^{5}-70 a^{4} b^{4} x^{4}+63 a^{5} b^{3} x^{3}+12705 a^{6} b^{2} x^{2}+21021 a^{7} b x +9009 a^{8}\right ) \sqrt {b \,x^{2}+a x}}{153153 a^{6} x^{9}}\)	\(105\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-256 b^{8} x^{8}+128 a \,b^{7} x^{7}-96 a^{2} b^{6} x^{6}+80 a^{3} b^{5} x^{5}-70 a^{4} b^{4} x^{4}+63 a^{5} b^{3} x^{3}+12705 a^{6} b^{2} x^{2}+21021 a^{7} b x +9009 a^{8}\right )}{153153 x^{8} \sqrt {x \left (b x +a \right )}\, a^{6}}\)	\(108\)
default	\(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{17 a \,x^{12}}-\frac {10 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{15 a \,x^{11}}-\frac {8 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{13 a \,x^{10}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{11 a \,x^{9}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{9 a \,x^{8}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a*x)^(5/2)/x^12,x,method=_RETURNVERBOSE)

[Out]

-2/17*(b*x^2+a*x)^(7/2)/a/x^12-10/17*b/a*(-2/15*(b*x^2+a*x)^(7/2)/a/x^11-8/15*b/a*(-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 [A]
time = 0.30, size = 222, normalized size = 1.46 \begin {gather*} \frac {512 \, \sqrt {b x^{2} + a x} b^{8}}{153153 \, a^{6} x} - \frac {256 \, \sqrt {b x^{2} + a x} b^{7}}{153153 \, a^{5} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} b^{6}}{51051 \, a^{4} x^{3}} - \frac {160 \, \sqrt {b x^{2} + a x} b^{5}}{153153 \, a^{3} x^{4}} + \frac {20 \, \sqrt {b x^{2} + a x} b^{4}}{21879 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{2431 \, a x^{6}} + \frac {\sqrt {b x^{2} + a x} b^{2}}{1326 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} a b}{1428 \, x^{8}} - \frac {5 \, \sqrt {b x^{2} + a x} a^{2}}{476 \, x^{9}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{84 \, x^{10}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{6 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^12,x, algorithm="maxima")

[Out]

512/153153*sqrt(b*x^2 + a*x)*b^8/(a^6*x) - 256/153153*sqrt(b*x^2 + a*x)*b^7/(a^5*x^2) + 64/51051*sqrt(b*x^2 +
a*x)*b^6/(a^4*x^3) - 160/153153*sqrt(b*x^2 + a*x)*b^5/(a^3*x^4) + 20/21879*sqrt(b*x^2 + a*x)*b^4/(a^2*x^5) - 2
/2431*sqrt(b*x^2 + a*x)*b^3/(a*x^6) + 1/1326*sqrt(b*x^2 + a*x)*b^2/x^7 - 1/1428*sqrt(b*x^2 + a*x)*a*b/x^8 - 5/
476*sqrt(b*x^2 + a*x)*a^2/x^9 + 5/84*(b*x^2 + a*x)^(3/2)*a/x^10 - 1/6*(b*x^2 + a*x)^(5/2)/x^11

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Fricas [A]
time = 1.48, size = 104, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (256 \, b^{8} x^{8} - 128 \, a b^{7} x^{7} + 96 \, a^{2} b^{6} x^{6} - 80 \, a^{3} b^{5} x^{5} + 70 \, a^{4} b^{4} x^{4} - 63 \, a^{5} b^{3} x^{3} - 12705 \, a^{6} b^{2} x^{2} - 21021 \, a^{7} b x - 9009 \, a^{8}\right )} \sqrt {b x^{2} + a x}}{153153 \, a^{6} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^12,x, algorithm="fricas")

[Out]

2/153153*(256*b^8*x^8 - 128*a*b^7*x^7 + 96*a^2*b^6*x^6 - 80*a^3*b^5*x^5 + 70*a^4*b^4*x^4 - 63*a^5*b^3*x^3 - 12
705*a^6*b^2*x^2 - 21021*a^7*b*x - 9009*a^8)*sqrt(b*x^2 + a*x)/(a^6*x^9)

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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^{12}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**12,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**12, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (128) = 256\).
time = 1.46, size = 339, normalized size = 2.23 \begin {gather*} \frac {2 \, {\left (816816 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{11} b^{\frac {11}{2}} + 5951088 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{10} a b^{5} + 19909890 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} a^{2} b^{\frac {9}{2}} + 40160120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a^{3} b^{4} + 54063009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{4} b^{\frac {7}{2}} + 50860719 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{5} b^{3} + 34051017 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{6} b^{\frac {5}{2}} + 16198875 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{7} b^{2} + 5360355 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{8} b^{\frac {3}{2}} + 1174173 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{9} b + 153153 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{10} \sqrt {b} + 9009 \, a^{11}\right )}}{153153 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^12,x, algorithm="giac")

[Out]

2/153153*(816816*(sqrt(b)*x - sqrt(b*x^2 + a*x))^11*b^(11/2) + 5951088*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*a*b^
5 + 19909890*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a^2*b^(9/2) + 40160120*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^3*b^
4 + 54063009*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^4*b^(7/2) + 50860719*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^5*b^
3 + 34051017*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^6*b^(5/2) + 16198875*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^7*b^
2 + 5360355*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^8*b^(3/2) + 1174173*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^9*b +
153153*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^10*sqrt(b) + 9009*a^11)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^17

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Mupad [B]
time = 2.19, size = 189, normalized size = 1.24 \begin {gather*} \frac {20\,b^4\,\sqrt {b\,x^2+a\,x}}{21879\,a^2\,x^5}-\frac {110\,b^2\,\sqrt {b\,x^2+a\,x}}{663\,x^7}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{2431\,a\,x^6}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{17\,x^9}-\frac {160\,b^5\,\sqrt {b\,x^2+a\,x}}{153153\,a^3\,x^4}+\frac {64\,b^6\,\sqrt {b\,x^2+a\,x}}{51051\,a^4\,x^3}-\frac {256\,b^7\,\sqrt {b\,x^2+a\,x}}{153153\,a^5\,x^2}+\frac {512\,b^8\,\sqrt {b\,x^2+a\,x}}{153153\,a^6\,x}-\frac {14\,a\,b\,\sqrt {b\,x^2+a\,x}}{51\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^2)^(5/2)/x^12,x)

[Out]

(20*b^4*(a*x + b*x^2)^(1/2))/(21879*a^2*x^5) - (110*b^2*(a*x + b*x^2)^(1/2))/(663*x^7) - (2*b^3*(a*x + b*x^2)^
(1/2))/(2431*a*x^6) - (2*a^2*(a*x + b*x^2)^(1/2))/(17*x^9) - (160*b^5*(a*x + b*x^2)^(1/2))/(153153*a^3*x^4) +
(64*b^6*(a*x + b*x^2)^(1/2))/(51051*a^4*x^3) - (256*b^7*(a*x + b*x^2)^(1/2))/(153153*a^5*x^2) + (512*b^8*(a*x
+ b*x^2)^(1/2))/(153153*a^6*x) - (14*a*b*(a*x + b*x^2)^(1/2))/(51*x^8)

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