3.1.80 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{17}} \, dx\) [80]

Optimal. Leaf size=251 \[ -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]

[Out]

-1/16*a^5*((b*x^3+a)^2)^(1/2)/x^16/(b*x^3+a)-5/13*a^4*b*((b*x^3+a)^2)^(1/2)/x^13/(b*x^3+a)-a^3*b^2*((b*x^3+a)^
2)^(1/2)/x^10/(b*x^3+a)-10/7*a^2*b^3*((b*x^3+a)^2)^(1/2)/x^7/(b*x^3+a)-5/4*a*b^4*((b*x^3+a)^2)^(1/2)/x^4/(b*x^
3+a)-b^5*((b*x^3+a)^2)^(1/2)/x/(b*x^3+a)

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Rubi [A]
time = 0.04, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 276} \begin {gather*} -\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^17,x]

[Out]

-1/16*(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^16*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(13
*x^13*(a + b*x^3)) - (a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^10*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a
*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*x^4*(a + b*x^3)) - (b^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3))

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{17}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{17}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^5 b^5}{x^{17}}+\frac {5 a^4 b^6}{x^{14}}+\frac {10 a^3 b^7}{x^{11}}+\frac {10 a^2 b^8}{x^8}+\frac {5 a b^9}{x^5}+\frac {b^{10}}{x^2}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 83, normalized size = 0.33 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (91 a^5+560 a^4 b x^3+1456 a^3 b^2 x^6+2080 a^2 b^3 x^9+1820 a b^4 x^{12}+1456 b^5 x^{15}\right )}{1456 x^{16} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^17,x]

[Out]

-1/1456*(Sqrt[(a + b*x^3)^2]*(91*a^5 + 560*a^4*b*x^3 + 1456*a^3*b^2*x^6 + 2080*a^2*b^3*x^9 + 1820*a*b^4*x^12 +
1456*b^5*x^15))/(x^16*(a + b*x^3))

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Maple [A]
time = 0.02, size = 80, normalized size = 0.32

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {1}{16} a^{5}-\frac {5}{13} a^{4} b \,x^{3}-b^{2} a^{3} x^{6}-\frac {10}{7} a^{2} b^{3} x^{9}-\frac {5}{4} b^{4} a \,x^{12}-b^{5} x^{15}\right )}{\left (b \,x^{3}+a \right ) x^{16}}\)	\(79\)
gosper	\(-\frac {\left (1456 b^{5} x^{15}+1820 b^{4} a \,x^{12}+2080 a^{2} b^{3} x^{9}+1456 b^{2} a^{3} x^{6}+560 a^{4} b \,x^{3}+91 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{1456 x^{16} \left (b \,x^{3}+a \right )^{5}}\)	\(80\)
default	\(-\frac {\left (1456 b^{5} x^{15}+1820 b^{4} a \,x^{12}+2080 a^{2} b^{3} x^{9}+1456 b^{2} a^{3} x^{6}+560 a^{4} b \,x^{3}+91 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{1456 x^{16} \left (b \,x^{3}+a \right )^{5}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^17,x,method=_RETURNVERBOSE)

[Out]

-1/1456*(1456*b^5*x^15+1820*a*b^4*x^12+2080*a^2*b^3*x^9+1456*a^3*b^2*x^6+560*a^4*b*x^3+91*a^5)*((b*x^3+a)^2)^(
5/2)/x^16/(b*x^3+a)^5

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Maxima [A]
time = 0.28, size = 59, normalized size = 0.24 \begin {gather*} -\frac {1456 \, b^{5} x^{15} + 1820 \, a b^{4} x^{12} + 2080 \, a^{2} b^{3} x^{9} + 1456 \, a^{3} b^{2} x^{6} + 560 \, a^{4} b x^{3} + 91 \, a^{5}}{1456 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^17,x, algorithm="maxima")

[Out]

-1/1456*(1456*b^5*x^15 + 1820*a*b^4*x^12 + 2080*a^2*b^3*x^9 + 1456*a^3*b^2*x^6 + 560*a^4*b*x^3 + 91*a^5)/x^16

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Fricas [A]
time = 0.36, size = 59, normalized size = 0.24 \begin {gather*} -\frac {1456 \, b^{5} x^{15} + 1820 \, a b^{4} x^{12} + 2080 \, a^{2} b^{3} x^{9} + 1456 \, a^{3} b^{2} x^{6} + 560 \, a^{4} b x^{3} + 91 \, a^{5}}{1456 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^17,x, algorithm="fricas")

[Out]

-1/1456*(1456*b^5*x^15 + 1820*a*b^4*x^12 + 2080*a^2*b^3*x^9 + 1456*a^3*b^2*x^6 + 560*a^4*b*x^3 + 91*a^5)/x^16

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{17}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**17,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**17, x)

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Giac [A]
time = 4.30, size = 107, normalized size = 0.43 \begin {gather*} -\frac {1456 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 1820 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 2080 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 1456 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 560 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 91 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{1456 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^17,x, algorithm="giac")

[Out]

-1/1456*(1456*b^5*x^15*sgn(b*x^3 + a) + 1820*a*b^4*x^12*sgn(b*x^3 + a) + 2080*a^2*b^3*x^9*sgn(b*x^3 + a) + 145
6*a^3*b^2*x^6*sgn(b*x^3 + a) + 560*a^4*b*x^3*sgn(b*x^3 + a) + 91*a^5*sgn(b*x^3 + a))/x^16

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Mupad [B]
time = 1.26, size = 231, normalized size = 0.92 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{16\,x^{16}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{4\,x^4\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{13\,x^{13}\,\left (b\,x^3+a\right )}-\frac {10\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^7\,\left (b\,x^3+a\right )}-\frac {a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^{10}\,\left (b\,x^3+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^17,x)

[Out]

- (a^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(16*x^16*(a + b*x^3)) - (b^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(x*(
a + b*x^3)) - (5*a*b^4*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(4*x^4*(a + b*x^3)) - (5*a^4*b*(a^2 + b^2*x^6 + 2*a*
b*x^3)^(1/2))/(13*x^13*(a + b*x^3)) - (10*a^2*b^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(7*x^7*(a + b*x^3)) - (a^
3*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(x^10*(a + b*x^3))

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