3.10.73 \(\int \frac {1}{(c x)^{7/2} (a+b x^2)^{3/4}} \, dx\) [973]

Optimal. Leaf size=55 \[ -\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}}+\frac {8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}} \]

[Out]

-2*(b*x^2+a)^(1/4)/a/c/(c*x)^(5/2)+8/5*(b*x^2+a)^(5/4)/a^2/c/(c*x)^(5/2)

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Rubi [A]
time = 0.01, antiderivative size = 55, 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 = {279, 270} \begin {gather*} \frac {8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(7/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*(a + b*x^2)^(1/4))/(a*c*(c*x)^(5/2)) + (8*(a + b*x^2)^(5/4))/(5*a^2*c*(c*x)^(5/2))

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 {1}{(c x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}}-\frac {4 \int \frac {\sqrt [4]{a+b x^2}}{(c x)^{7/2}} \, dx}{a}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}}+\frac {8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 34, normalized size = 0.62 \begin {gather*} -\frac {2 x \left (a-4 b x^2\right ) \sqrt [4]{a+b x^2}}{5 a^2 (c x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*x*(a - 4*b*x^2)*(a + b*x^2)^(1/4))/(5*a^2*(c*x)^(7/2))

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Maple [A]
time = 0.12, size = 29, normalized size = 0.53

method	result	size
gosper	\(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-4 b \,x^{2}+a \right )}{5 a^{2} \left (c x \right )^{\frac {7}{2}}}\)	\(29\)
risch	\(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-4 b \,x^{2}+a \right )}{5 c^{3} \sqrt {c x}\, a^{2} x^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(7/2)/(b*x^2+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-2/5*x*(b*x^2+a)^(1/4)*(-4*b*x^2+a)/a^2/(c*x)^(7/2)

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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(1/(c*x)^(7/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(7/2)), x)

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Fricas [A]
time = 1.30, size = 35, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (4 \, b x^{2} - a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{5 \, a^{2} c^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(7/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

2/5*(4*b*x^2 - a)*(b*x^2 + a)^(1/4)*sqrt(c*x)/(a^2*c^4*x^3)

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Sympy [A]
time = 17.07, size = 78, normalized size = 1.42 \begin {gather*} - \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{8 a c^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} + \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} c^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(7/2)/(b*x**2+a)**(3/4),x)

[Out]

-b**(1/4)*(a/(b*x**2) + 1)**(1/4)*gamma(-5/4)/(8*a*c**(7/2)*x**2*gamma(3/4)) + b**(5/4)*(a/(b*x**2) + 1)**(1/4
)*gamma(-5/4)/(2*a**2*c**(7/2)*gamma(3/4))

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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(1/(c*x)^(7/2)/(b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(7/2)), x)

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Mupad [B]
time = 4.98, size = 40, normalized size = 0.73 \begin {gather*} -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2}{5\,a\,c^3}-\frac {8\,b\,x^2}{5\,a^2\,c^3}\right )}{x^2\,\sqrt {c\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(7/2)*(a + b*x^2)^(3/4)),x)

[Out]

-((a + b*x^2)^(1/4)*(2/(5*a*c^3) - (8*b*x^2)/(5*a^2*c^3)))/(x^2*(c*x)^(1/2))

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