3.6.23 \(\int \frac {x^4}{(a+b x^2)^{9/2}} \, dx\) [523]

Optimal. Leaf size=44 \[ \frac {x^5}{5 a \left (a+b x^2\right )^{7/2}}+\frac {2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} \frac {2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}+\frac {x^5}{5 a \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^2)^(9/2),x]

[Out]

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

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 277

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

Rubi steps

\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac {x^5}{5 a \left (a+b x^2\right )^{7/2}}+\frac {(2 b) \int \frac {x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{5 a}\\ &=\frac {x^5}{5 a \left (a+b x^2\right )^{7/2}}+\frac {2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 31, normalized size = 0.70 \begin {gather*} \frac {7 a x^5+2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^2)^(9/2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(36)=72\).
time = 0.04, size = 120, normalized size = 2.73

method	result	size
gosper	\(\frac {x^{5} \left (2 b \,x^{2}+7 a \right )}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2}}\)	\(28\)
trager	\(\frac {x^{5} \left (2 b \,x^{2}+7 a \right )}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2}}\)	\(28\)
default	\(-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (36) = 72\).
time = 0.31, size = 85, normalized size = 1.93 \begin {gather*} -\frac {x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {3 \, x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/4*x^3/((b*x^2 + a)^(7/2)*b) + 3/140*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*x/(sqrt(b*x^2 + a)*a^2*b^2) + 1/35*x/(
(b*x^2 + a)^(3/2)*a*b^2) - 3/28*a*x/((b*x^2 + a)^(7/2)*b^2)

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Fricas [A]
time = 1.23, size = 71, normalized size = 1.61 \begin {gather*} \frac {{\left (2 \, b x^{7} + 7 \, a x^{5}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{2} b^{4} x^{8} + 4 \, a^{3} b^{3} x^{6} + 6 \, a^{4} b^{2} x^{4} + 4 \, a^{5} b x^{2} + a^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/35*(2*b*x^7 + 7*a*x^5)*sqrt(b*x^2 + a)/(a^2*b^4*x^8 + 4*a^3*b^3*x^6 + 6*a^4*b^2*x^4 + 4*a^5*b*x^2 + a^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (37) = 74\).
time = 0.78, size = 199, normalized size = 4.52 \begin {gather*} \frac {7 a x^{5}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{7}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**2+a)**(9/2),x)

[Out]

7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sq
rt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a)) + 2*b*x**7/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105
*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1
+ b*x**2/a))

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Giac [A]
time = 0.54, size = 29, normalized size = 0.66 \begin {gather*} \frac {x^{5} {\left (\frac {2 \, b x^{2}}{a^{2}} + \frac {7}{a}\right )}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.88, size = 68, normalized size = 1.55 \begin {gather*} \frac {2\,x}{35\,a^2\,b^2\,\sqrt {b\,x^2+a}}-\frac {8\,x}{35\,b^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {x}{35\,a\,b^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {a\,x}{7\,b^2\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x^2)^(9/2),x)

[Out]

(2*x)/(35*a^2*b^2*(a + b*x^2)^(1/2)) - (8*x)/(35*b^2*(a + b*x^2)^(5/2)) + x/(35*a*b^2*(a + b*x^2)^(3/2)) + (a*
x)/(7*b^2*(a + b*x^2)^(7/2))

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