3.3.62 \(\int \frac {x^4 (c+d x^2)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac {\sqrt {a} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {a x (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac {x (b c-2 a d)}{b^3}+\frac {d x^3}{3 b^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 1153, 205} \begin {gather*} \frac {a x (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac {x (b c-2 a d)}{b^3}-\frac {\sqrt {a} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {d x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - 2*a*d)*x)/b^3 + (d*x^3)/(3*b^2) + (a*(b*c - a*d)*x)/(2*b^3*(a + b*x^2)) - (Sqrt[a]*(3*b*c - 5*a*d)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 89, normalized size = 1.02 \begin {gather*} \frac {x \left (a b c-a^2 d\right )}{2 b^3 \left (a+b x^2\right )}+\frac {\sqrt {a} (5 a d-3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {x (b c-2 a d)}{b^3}+\frac {d x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - 2*a*d)*x)/b^3 + (d*x^3)/(3*b^2) + ((a*b*c - a^2*d)*x)/(2*b^3*(a + b*x^2)) + (Sqrt[a]*(-3*b*c + 5*a*d)*
ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, size = 240, normalized size = 2.76 \begin {gather*} \left [\frac {4 \, b^{2} d x^{5} + 4 \, {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{3} - 3 \, {\left (3 \, a b c - 5 \, a^{2} d + {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, a b c - 5 \, a^{2} d\right )} x}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {2 \, b^{2} d x^{5} + 2 \, {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{3} - 3 \, {\left (3 \, a b c - 5 \, a^{2} d + {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, {\left (3 \, a b c - 5 \, a^{2} d\right )} x}{6 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(4*b^2*d*x^5 + 4*(3*b^2*c - 5*a*b*d)*x^3 - 3*(3*a*b*c - 5*a^2*d + (3*b^2*c - 5*a*b*d)*x^2)*sqrt(-a/b)*lo
g((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 6*(3*a*b*c - 5*a^2*d)*x)/(b^4*x^2 + a*b^3), 1/6*(2*b^2*d*x^5 +
2*(3*b^2*c - 5*a*b*d)*x^3 - 3*(3*a*b*c - 5*a^2*d + (3*b^2*c - 5*a*b*d)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a)
+ 3*(3*a*b*c - 5*a^2*d)*x)/(b^4*x^2 + a*b^3)]

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giac [A]  time = 0.39, size = 88, normalized size = 1.01 \begin {gather*} -\frac {{\left (3 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {a b c x - a^{2} d x}{2 \, {\left (b x^{2} + a\right )} b^{3}} + \frac {b^{4} d x^{3} + 3 \, b^{4} c x - 6 \, a b^{3} d x}{3 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*a*b*c - 5*a^2*d)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/2*(a*b*c*x - a^2*d*x)/((b*x^2 + a)*b^3) + 1
/3*(b^4*d*x^3 + 3*b^4*c*x - 6*a*b^3*d*x)/b^6

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maple [A]  time = 0.01, size = 105, normalized size = 1.21 \begin {gather*} \frac {d \,x^{3}}{3 b^{2}}-\frac {a^{2} d x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {a c x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {2 a d x}{b^{3}}+\frac {c x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*d*x^3/b^2-2/b^3*a*d*x+1/b^2*c*x-1/2*a^2/b^3*x/(b*x^2+a)*d+1/2*a/b^2*x/(b*x^2+a)*c+5/2*a^2/b^3/(a*b)^(1/2)*
arctan(1/(a*b)^(1/2)*b*x)*d-3/2*a/b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c

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maxima [A]  time = 2.24, size = 84, normalized size = 0.97 \begin {gather*} \frac {{\left (a b c - a^{2} d\right )} x}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} - \frac {{\left (3 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {b d x^{3} + 3 \, {\left (b c - 2 \, a d\right )} x}{3 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.08, size = 104, normalized size = 1.20 \begin {gather*} x\,\left (\frac {c}{b^2}-\frac {2\,a\,d}{b^3}\right )+\frac {d\,x^3}{3\,b^2}-\frac {x\,\left (\frac {a^2\,d}{2}-\frac {a\,b\,c}{2}\right )}{b^4\,x^2+a\,b^3}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (5\,a\,d-3\,b\,c\right )}{5\,a^2\,d-3\,a\,b\,c}\right )\,\left (5\,a\,d-3\,b\,c\right )}{2\,b^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.63, size = 129, normalized size = 1.48 \begin {gather*} x \left (- \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{2} d + a b c\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {\sqrt {- \frac {a}{b^{7}}} \left (5 a d - 3 b c\right ) \log {\left (- b^{3} \sqrt {- \frac {a}{b^{7}}} + x \right )}}{4} + \frac {\sqrt {- \frac {a}{b^{7}}} \left (5 a d - 3 b c\right ) \log {\left (b^{3} \sqrt {- \frac {a}{b^{7}}} + x \right )}}{4} + \frac {d x^{3}}{3 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-2*a*d/b**3 + c/b**2) + x*(-a**2*d + a*b*c)/(2*a*b**3 + 2*b**4*x**2) - sqrt(-a/b**7)*(5*a*d - 3*b*c)*log(-b
**3*sqrt(-a/b**7) + x)/4 + sqrt(-a/b**7)*(5*a*d - 3*b*c)*log(b**3*sqrt(-a/b**7) + x)/4 + d*x**3/(3*b**2)

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