3.4.69 \(\int x^4 (a+b x^2)^{3/2} \, dx\)

Optimal. Leaf size=115 \[ \frac {3 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}-\frac {3 a^3 x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 x^3 \sqrt {a+b x^2}}{64 b}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{16} a x^5 \sqrt {a+b x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \begin {gather*} -\frac {3 a^3 x \sqrt {a+b x^2}}{128 b^2}+\frac {3 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}+\frac {a^2 x^3 \sqrt {a+b x^2}}{64 b}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{16} a x^5 \sqrt {a+b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^(3/2),x]

[Out]

(-3*a^3*x*Sqrt[a + b*x^2])/(128*b^2) + (a^2*x^3*Sqrt[a + b*x^2])/(64*b) + (a*x^5*Sqrt[a + b*x^2])/16 + (x^5*(a
+ b*x^2)^(3/2))/8 + (3*a^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(5/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 279

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

Rule 321

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

Rubi steps

\begin {align*} \int x^4 \left (a+b x^2\right )^{3/2} \, dx &=\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{8} (3 a) \int x^4 \sqrt {a+b x^2} \, dx\\ &=\frac {1}{16} a x^5 \sqrt {a+b x^2}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{16} a^2 \int \frac {x^4}{\sqrt {a+b x^2}} \, dx\\ &=\frac {a^2 x^3 \sqrt {a+b x^2}}{64 b}+\frac {1}{16} a x^5 \sqrt {a+b x^2}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}-\frac {\left (3 a^3\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{64 b}\\ &=-\frac {3 a^3 x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 x^3 \sqrt {a+b x^2}}{64 b}+\frac {1}{16} a x^5 \sqrt {a+b x^2}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {\left (3 a^4\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^2}\\ &=-\frac {3 a^3 x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 x^3 \sqrt {a+b x^2}}{64 b}+\frac {1}{16} a x^5 \sqrt {a+b x^2}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {\left (3 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^2}\\ &=-\frac {3 a^3 x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 x^3 \sqrt {a+b x^2}}{64 b}+\frac {1}{16} a x^5 \sqrt {a+b x^2}+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac {3 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 94, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a+b x^2} \left (\frac {3 a^{7/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}+\sqrt {b} x \left (-3 a^3+2 a^2 b x^2+24 a b^2 x^4+16 b^3 x^6\right )\right )}{128 b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^(3/2),x]

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-3*a^3 + 2*a^2*b*x^2 + 24*a*b^2*x^4 + 16*b^3*x^6) + (3*a^(7/2)*ArcSinh[(Sqrt[b]*x
)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(128*b^(5/2))

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IntegrateAlgebraic [A]  time = 0.08, size = 85, normalized size = 0.74 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-3 a^3 x+2 a^2 b x^3+24 a b^2 x^5+16 b^3 x^7\right )}{128 b^2}-\frac {3 a^4 \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{128 b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4*(a + b*x^2)^(3/2),x]

[Out]

(Sqrt[a + b*x^2]*(-3*a^3*x + 2*a^2*b*x^3 + 24*a*b^2*x^5 + 16*b^3*x^7))/(128*b^2) - (3*a^4*Log[-(Sqrt[b]*x) + S
qrt[a + b*x^2]])/(128*b^(5/2))

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fricas [A]  time = 0.93, size = 168, normalized size = 1.46 \begin {gather*} \left [\frac {3 \, a^{4} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (16 \, b^{4} x^{7} + 24 \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} - 3 \, a^{3} b x\right )} \sqrt {b x^{2} + a}}{256 \, b^{3}}, -\frac {3 \, a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (16 \, b^{4} x^{7} + 24 \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} - 3 \, a^{3} b x\right )} \sqrt {b x^{2} + a}}{128 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/256*(3*a^4*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(16*b^4*x^7 + 24*a*b^3*x^5 + 2*a^2*b
^2*x^3 - 3*a^3*b*x)*sqrt(b*x^2 + a))/b^3, -1/128*(3*a^4*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (16*b^4*
x^7 + 24*a*b^3*x^5 + 2*a^2*b^2*x^3 - 3*a^3*b*x)*sqrt(b*x^2 + a))/b^3]

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giac [A]  time = 0.66, size = 76, normalized size = 0.66 \begin {gather*} \frac {1}{128} \, {\left (2 \, {\left (4 \, {\left (2 \, b x^{2} + 3 \, a\right )} x^{2} + \frac {a^{2}}{b}\right )} x^{2} - \frac {3 \, a^{3}}{b^{2}}\right )} \sqrt {b x^{2} + a} x - \frac {3 \, a^{4} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(2*(4*(2*b*x^2 + 3*a)*x^2 + a^2/b)*x^2 - 3*a^3/b^2)*sqrt(b*x^2 + a)*x - 3/128*a^4*log(abs(-sqrt(b)*x + s
qrt(b*x^2 + a)))/b^(5/2)

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maple [A]  time = 0.01, size = 95, normalized size = 0.83 \begin {gather*} \frac {3 a^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}+\frac {3 \sqrt {b \,x^{2}+a}\, a^{3} x}{128 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} x^{3}}{8 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} x}{64 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a x}{16 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^3*(b*x^2+a)^(5/2)/b-1/16*a/b^2*x*(b*x^2+a)^(5/2)+1/64*a^2/b^2*x*(b*x^2+a)^(3/2)+3/128*a^3*x*(b*x^2+a)^(1
/2)/b^2+3/128*a^4/b^(5/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.36, size = 87, normalized size = 0.76 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{3}}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{3} x}{128 \, b^{2}} + \frac {3 \, a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^2 + a)^(5/2)*x^3/b - 1/16*(b*x^2 + a)^(5/2)*a*x/b^2 + 1/64*(b*x^2 + a)^(3/2)*a^2*x/b^2 + 3/128*sqrt(b
*x^2 + a)*a^3*x/b^2 + 3/128*a^4*arcsinh(b*x/sqrt(a*b))/b^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^(3/2),x)

[Out]

int(x^4*(a + b*x^2)^(3/2), x)

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sympy [A]  time = 8.20, size = 148, normalized size = 1.29 \begin {gather*} - \frac {3 a^{\frac {7}{2}} x}{128 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {5}{2}} x^{3}}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {13 a^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 \sqrt {a} b x^{7}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} + \frac {b^{2} x^{9}}{8 \sqrt {a} \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)**(3/2),x)

[Out]

-3*a**(7/2)*x/(128*b**2*sqrt(1 + b*x**2/a)) - a**(5/2)*x**3/(128*b*sqrt(1 + b*x**2/a)) + 13*a**(3/2)*x**5/(64*
sqrt(1 + b*x**2/a)) + 5*sqrt(a)*b*x**7/(16*sqrt(1 + b*x**2/a)) + 3*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(5/2)
) + b**2*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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