3.1.82 \(\int \frac {x^{15/2}}{(a x+b x^3)^{9/2}} \, dx\) [82]

Optimal. Leaf size=51 \[ -\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}} \]

[Out]

-1/7*x^(11/2)/b/(b*x^3+a*x)^(7/2)-2/35*x^(5/2)/b^2/(b*x^3+a*x)^(5/2)

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Rubi [A]
time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2040, 2039} \begin {gather*} -\frac {2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(15/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-1/7*x^(11/2)/(b*(a*x + b*x^3)^(7/2)) - (2*x^(5/2))/(35*b^2*(a*x + b*x^3)^(5/2))

Rule 2039

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-c^(j - 1))*(c*x)^(m - j
+ 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2040

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

Rubi steps

\begin {align*} \int \frac {x^{15/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac {2 \int \frac {x^{9/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 35, normalized size = 0.69 \begin {gather*} \frac {x^{7/2} \left (-2 a-7 b x^2\right )}{35 b^2 \left (x \left (a+b x^2\right )\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(15/2)/(a*x + b*x^3)^(9/2),x]

[Out]

(x^(7/2)*(-2*a - 7*b*x^2))/(35*b^2*(x*(a + b*x^2))^(7/2))

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Maple [A]
time = 0.36, size = 39, normalized size = 0.76

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right ) \left (7 b \,x^{2}+2 a \right ) x^{\frac {9}{2}}}{35 b^{2} \left (b \,x^{3}+a x \right )^{\frac {9}{2}}}\)	\(37\)
default	\(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (7 b \,x^{2}+2 a \right )}{35 \sqrt {x}\, \left (b \,x^{2}+a \right )^{4} b^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(15/2)/(b*x^3+a*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/35/x^(1/2)*(x*(b*x^2+a))^(1/2)*(7*b*x^2+2*a)/(b*x^2+a)^4/b^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(15/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(15/2)/(b*x^3 + a*x)^(9/2), x)

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Fricas [A]
time = 1.75, size = 75, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {b x^{3} + a x} {\left (7 \, b x^{2} + 2 \, a\right )} \sqrt {x}}{35 \, {\left (b^{6} x^{9} + 4 \, a b^{5} x^{7} + 6 \, a^{2} b^{4} x^{5} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(15/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")

[Out]

-1/35*sqrt(b*x^3 + a*x)*(7*b*x^2 + 2*a)*sqrt(x)/(b^6*x^9 + 4*a*b^5*x^7 + 6*a^2*b^4*x^5 + 4*a^3*b^3*x^3 + a^4*b
^2*x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(15/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5457 deep

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Giac [A]
time = 1.62, size = 33, normalized size = 0.65 \begin {gather*} -\frac {7 \, b x^{2} + 2 \, a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {2}{35 \, a^{\frac {5}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(15/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")

[Out]

-1/35*(7*b*x^2 + 2*a)/((b*x^2 + a)^(7/2)*b^2) + 2/35/(a^(5/2)*b^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{15/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(15/2)/(a*x + b*x^3)^(9/2),x)

[Out]

int(x^(15/2)/(a*x + b*x^3)^(9/2), x)

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