3.8.61 \(\int \frac {(a+b x^2)^{4/3}}{(c x)^{29/3}} \, dx\) [761]

Optimal. Leaf size=85 \[ -\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}+\frac {9 \left (a+b x^2\right )^{10/3}}{70 a^2 c (c x)^{26/3}}-\frac {27 \left (a+b x^2\right )^{13/3}}{910 a^3 c (c x)^{26/3}} \]

[Out]

-3/14*(b*x^2+a)^(7/3)/a/c/(c*x)^(26/3)+9/70*(b*x^2+a)^(10/3)/a^2/c/(c*x)^(26/3)-27/910*(b*x^2+a)^(13/3)/a^3/c/
(c*x)^(26/3)

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Rubi [A]
time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {279, 270} \begin {gather*} -\frac {27 \left (a+b x^2\right )^{13/3}}{910 a^3 c (c x)^{26/3}}+\frac {9 \left (a+b x^2\right )^{10/3}}{70 a^2 c (c x)^{26/3}}-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(4/3)/(c*x)^(29/3),x]

[Out]

(-3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(26/3)) + (9*(a + b*x^2)^(10/3))/(70*a^2*c*(c*x)^(26/3)) - (27*(a + b*x^2
)^(13/3))/(910*a^3*c*(c*x)^(26/3))

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 {\left (a+b x^2\right )^{4/3}}{(c x)^{29/3}} \, dx &=-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}-\frac {6 \int \frac {\left (a+b x^2\right )^{7/3}}{(c x)^{29/3}} \, dx}{7 a}\\ &=-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}+\frac {9 \left (a+b x^2\right )^{10/3}}{70 a^2 c (c x)^{26/3}}+\frac {9 \int \frac {\left (a+b x^2\right )^{10/3}}{(c x)^{29/3}} \, dx}{35 a^2}\\ &=-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}+\frac {9 \left (a+b x^2\right )^{10/3}}{70 a^2 c (c x)^{26/3}}-\frac {27 \left (a+b x^2\right )^{13/3}}{910 a^3 c (c x)^{26/3}}\\ \end {align*}

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Mathematica [A]
time = 3.82, size = 47, normalized size = 0.55 \begin {gather*} -\frac {3 x \left (a+b x^2\right )^{7/3} \left (35 a^2-21 a b x^2+9 b^2 x^4\right )}{910 a^3 (c x)^{29/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(4/3)/(c*x)^(29/3),x]

[Out]

(-3*x*(a + b*x^2)^(7/3)*(35*a^2 - 21*a*b*x^2 + 9*b^2*x^4))/(910*a^3*(c*x)^(29/3))

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Maple [A]
time = 0.05, size = 42, normalized size = 0.49

method	result	size
gosper	\(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {7}{3}} \left (9 b^{2} x^{4}-21 a b \,x^{2}+35 a^{2}\right )}{910 a^{3} \left (c x \right )^{\frac {29}{3}}}\)	\(42\)
risch	\(-\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (9 b^{4} x^{8}-3 a \,b^{3} x^{6}+2 a^{2} b^{2} x^{4}+49 a^{3} b \,x^{2}+35 a^{4}\right )}{910 c^{9} \left (c x \right )^{\frac {2}{3}} x^{8} a^{3}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(4/3)/(c*x)^(29/3),x,method=_RETURNVERBOSE)

[Out]

-3/910*x*(b*x^2+a)^(7/3)*(9*b^2*x^4-21*a*b*x^2+35*a^2)/a^3/(c*x)^(29/3)

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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)^(4/3)/(c*x)^(29/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(29/3), x)

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Fricas [A]
time = 1.02, size = 68, normalized size = 0.80 \begin {gather*} -\frac {3 \, {\left (9 \, b^{4} x^{8} - 3 \, a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{4} + 49 \, a^{3} b x^{2} + 35 \, a^{4}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{910 \, a^{3} c^{10} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(29/3),x, algorithm="fricas")

[Out]

-3/910*(9*b^4*x^8 - 3*a*b^3*x^6 + 2*a^2*b^2*x^4 + 49*a^3*b*x^2 + 35*a^4)*(b*x^2 + a)^(1/3)*(c*x)^(1/3)/(a^3*c^
10*x^9)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(4/3)/(c*x)**(29/3),x)

[Out]

Timed out

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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)^(4/3)/(c*x)^(29/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(29/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{4/3}}{{\left (c\,x\right )}^{29/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(4/3)/(c*x)^(29/3),x)

[Out]

int((a + b*x^2)^(4/3)/(c*x)^(29/3), x)

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