3.1.90 \(\int \frac {x^7 (A+B x^2)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=109 \[ \frac {a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac {a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac {3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac {x^2 (A b-3 a B)}{2 b^4}+\frac {B x^4}{4 b^3} \]

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Rubi [A]  time = 0.12, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} -\frac {a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}+\frac {a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}+\frac {x^2 (A b-3 a B)}{2 b^4}-\frac {3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac {B x^4}{4 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

((A*b - 3*a*B)*x^2)/(2*b^4) + (B*x^4)/(4*b^3) + (a^3*(A*b - a*B))/(4*b^5*(a + b*x^2)^2) - (a^2*(3*A*b - 4*a*B)
)/(2*b^5*(a + b*x^2)) - (3*a*(A*b - 2*a*B)*Log[a + b*x^2])/(2*b^5)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A b-3 a B}{b^4}+\frac {B x}{b^3}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^3}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^2}+\frac {3 a (-A b+2 a B)}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {(A b-3 a B) x^2}{2 b^4}+\frac {B x^4}{4 b^3}+\frac {a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac {a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac {3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 94, normalized size = 0.86 \begin {gather*} \frac {\frac {a^3 (A b-a B)}{\left (a+b x^2\right )^2}+\frac {2 a^2 (4 a B-3 A b)}{a+b x^2}+2 b x^2 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^2\right )+b^2 B x^4}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(2*b*(A*b - 3*a*B)*x^2 + b^2*B*x^4 + (a^3*(A*b - a*B))/(a + b*x^2)^2 + (2*a^2*(-3*A*b + 4*a*B))/(a + b*x^2) +
6*a*(-(A*b) + 2*a*B)*Log[a + b*x^2])/(4*b^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

IntegrateAlgebraic[(x^7*(A + B*x^2))/(a + b*x^2)^3, x]

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fricas [A]  time = 0.45, size = 179, normalized size = 1.64 \begin {gather*} \frac {B b^{4} x^{8} - 2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (2 \, B a^{4} - A a^{3} b + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{4} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/4*(B*b^4*x^8 - 2*(2*B*a*b^3 - A*b^4)*x^6 + 7*B*a^4 - 5*A*a^3*b - (11*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(B*a^3*b
- 2*A*a^2*b^2)*x^2 + 6*(2*B*a^4 - A*a^3*b + (2*B*a^2*b^2 - A*a*b^3)*x^4 + 2*(2*B*a^3*b - A*a^2*b^2)*x^2)*log(
b*x^2 + a))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5)

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giac [A]  time = 0.36, size = 132, normalized size = 1.21 \begin {gather*} \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac {B b^{3} x^{4} - 6 \, B a b^{2} x^{2} + 2 \, A b^{3} x^{2}}{4 \, b^{6}} - \frac {18 \, B a^{2} b^{2} x^{4} - 9 \, A a b^{3} x^{4} + 28 \, B a^{3} b x^{2} - 12 \, A a^{2} b^{2} x^{2} + 11 \, B a^{4} - 4 \, A a^{3} b}{4 \, {\left (b x^{2} + a\right )}^{2} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

3/2*(2*B*a^2 - A*a*b)*log(abs(b*x^2 + a))/b^5 + 1/4*(B*b^3*x^4 - 6*B*a*b^2*x^2 + 2*A*b^3*x^2)/b^6 - 1/4*(18*B*
a^2*b^2*x^4 - 9*A*a*b^3*x^4 + 28*B*a^3*b*x^2 - 12*A*a^2*b^2*x^2 + 11*B*a^4 - 4*A*a^3*b)/((b*x^2 + a)^2*b^5)

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maple [A]  time = 0.02, size = 134, normalized size = 1.23 \begin {gather*} \frac {B \,x^{4}}{4 b^{3}}+\frac {A \,a^{3}}{4 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {A \,x^{2}}{2 b^{3}}-\frac {B \,a^{4}}{4 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {3 B a \,x^{2}}{2 b^{4}}-\frac {3 A \,a^{2}}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {3 A a \ln \left (b \,x^{2}+a \right )}{2 b^{4}}+\frac {2 B \,a^{3}}{\left (b \,x^{2}+a \right ) b^{5}}+\frac {3 B \,a^{2} \ln \left (b \,x^{2}+a \right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

1/4*B*x^4/b^3-3/2/b^4*B*x^2*a+1/2/b^3*A*x^2-3/2*a^2/b^4/(b*x^2+a)*A+2*a^3/b^5/(b*x^2+a)*B+1/4*a^3/b^4/(b*x^2+a
)^2*A-1/4*a^4/b^5/(b*x^2+a)^2*B-3/2*a/b^4*ln(b*x^2+a)*A+3*a^2/b^5*ln(b*x^2+a)*B

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maxima [A]  time = 1.09, size = 116, normalized size = 1.06 \begin {gather*} \frac {7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}}{4 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac {B b x^{4} - 2 \, {\left (3 \, B a - A b\right )} x^{2}}{4 \, b^{4}} + \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 118, normalized size = 1.08 \begin {gather*} \frac {\frac {7\,B\,a^4-5\,A\,a^3\,b}{4\,b}+x^2\,\left (2\,B\,a^3-\frac {3\,A\,a^2\,b}{2}\right )}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+x^2\,\left (\frac {A}{2\,b^3}-\frac {3\,B\,a}{2\,b^4}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (6\,B\,a^2-3\,A\,a\,b\right )}{2\,b^5}+\frac {B\,x^4}{4\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(A + B*x^2))/(a + b*x^2)^3,x)

[Out]

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

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sympy [A]  time = 1.41, size = 119, normalized size = 1.09 \begin {gather*} \frac {B x^{4}}{4 b^{3}} + \frac {3 a \left (- A b + 2 B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{2} \left (\frac {A}{2 b^{3}} - \frac {3 B a}{2 b^{4}}\right ) + \frac {- 5 A a^{3} b + 7 B a^{4} + x^{2} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{4 a^{2} b^{5} + 8 a b^{6} x^{2} + 4 b^{7} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

B*x**4/(4*b**3) + 3*a*(-A*b + 2*B*a)*log(a + b*x**2)/(2*b**5) + x**2*(A/(2*b**3) - 3*B*a/(2*b**4)) + (-5*A*a**
3*b + 7*B*a**4 + x**2*(-6*A*a**2*b**2 + 8*B*a**3*b))/(4*a**2*b**5 + 8*a*b**6*x**2 + 4*b**7*x**4)

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