3.10.42 \(\int \frac {\sqrt [4]{a-b x^2}}{(c x)^{15/2}} \, dx\) [942]

Optimal. Leaf size=88 \[ -\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}} \]

[Out]

-2/5*(-b*x^2+a)^(5/4)/a/c/(c*x)^(13/2)+16/45*(-b*x^2+a)^(9/4)/a^2/c/(c*x)^(13/2)-64/585*(-b*x^2+a)^(13/4)/a^3/
c/(c*x)^(13/2)

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Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {279, 270} \begin {gather*} -\frac {64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(1/4)/(c*x)^(15/2),x]

[Out]

(-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(13/2)) + (16*(a - b*x^2)^(9/4))/(45*a^2*c*(c*x)^(13/2)) - (64*(a - b*x^2)
^(13/4))/(585*a^3*c*(c*x)^(13/2))

Rule 270

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

Rule 279

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

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{15/2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}-\frac {8 \int \frac {\left (a-b x^2\right )^{5/4}}{(c x)^{15/2}} \, dx}{5 a}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}+\frac {32 \int \frac {\left (a-b x^2\right )^{9/4}}{(c x)^{15/2}} \, dx}{45 a^2}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 48, normalized size = 0.55 \begin {gather*} -\frac {2 x \left (a-b x^2\right )^{5/4} \left (45 a^2+40 a b x^2+32 b^2 x^4\right )}{585 a^3 (c x)^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(1/4)/(c*x)^(15/2),x]

[Out]

(-2*x*(a - b*x^2)^(5/4)*(45*a^2 + 40*a*b*x^2 + 32*b^2*x^4))/(585*a^3*(c*x)^(15/2))

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Maple [A]
time = 0.06, size = 43, normalized size = 0.49

method	result	size
gosper	\(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{4}+40 a b \,x^{2}+45 a^{2}\right )}{585 a^{3} \left (c x \right )^{\frac {15}{2}}}\)	\(43\)
risch	\(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{2}+a \right )^{3}\right )^{\frac {1}{4}} \left (-32 b^{3} x^{6}-8 a \,b^{2} x^{4}-5 a^{2} b \,x^{2}+45 a^{3}\right )}{585 \sqrt {c x}\, \left (-\left (b \,x^{2}-a \right )^{3}\right )^{\frac {1}{4}} c^{7} x^{6} a^{3}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(1/4)/(c*x)^(15/2),x,method=_RETURNVERBOSE)

[Out]

-2/585*x*(-b*x^2+a)^(5/4)*(32*b^2*x^4+40*a*b*x^2+45*a^2)/a^3/(c*x)^(15/2)

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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((-b*x^2+a)^(1/4)/(c*x)^(15/2),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(15/2), x)

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Fricas [A]
time = 1.63, size = 58, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (32 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} - 45 \, a^{3}\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{585 \, a^{3} c^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(15/2),x, algorithm="fricas")

[Out]

2/585*(32*b^3*x^6 + 8*a*b^2*x^4 + 5*a^2*b*x^2 - 45*a^3)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a^3*c^8*x^7)

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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((-b*x**2+a)**(1/4)/(c*x)**(15/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4497 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(15/2),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(15/2), x)

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Mupad [B]
time = 4.91, size = 65, normalized size = 0.74 \begin {gather*} \frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2\,b\,x^2}{117\,a\,c^7}-\frac {2}{13\,c^7}+\frac {16\,b^2\,x^4}{585\,a^2\,c^7}+\frac {64\,b^3\,x^6}{585\,a^3\,c^7}\right )}{x^6\,\sqrt {c\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^2)^(1/4)/(c*x)^(15/2),x)

[Out]

((a - b*x^2)^(1/4)*((2*b*x^2)/(117*a*c^7) - 2/(13*c^7) + (16*b^2*x^4)/(585*a^2*c^7) + (64*b^3*x^6)/(585*a^3*c^
7)))/(x^6*(c*x)^(1/2))

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