3.35 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^3} \, dx\)

Optimal. Leaf size=113 \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{5}{2} a^3 b x^2 (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac{a^5 A}{2 x^2}+\frac{1}{8} b^4 x^8 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

[Out]

-(a^5*A)/(2*x^2) + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^6)
/6 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^10)/10 + a^4*(5*A*b + a*B)*Log[x]

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Rubi [A]  time = 0.112783, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{5}{2} a^3 b x^2 (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac{a^5 A}{2 x^2}+\frac{1}{8} b^4 x^8 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^3,x]

[Out]

-(a^5*A)/(2*x^2) + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^6)
/6 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^10)/10 + a^4*(5*A*b + a*B)*Log[x]

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]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (5 a^3 b (2 A b+a B)+\frac{a^5 A}{x^2}+\frac{a^4 (5 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+5 a b^3 (A b+2 a B) x^2+b^4 (A b+5 a B) x^3+b^5 B x^4\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 A}{2 x^2}+\frac{5}{2} a^3 b (2 A b+a B) x^2+\frac{5}{2} a^2 b^2 (A b+a B) x^4+\frac{5}{6} a b^3 (A b+2 a B) x^6+\frac{1}{8} b^4 (A b+5 a B) x^8+\frac{1}{10} b^5 B x^{10}+a^4 (5 A b+a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0418384, size = 115, normalized size = 1.02 \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{5}{2} a^3 b x^2 (a B+2 A b)+\log (x) \left (5 a^4 A b+a^5 B\right )-\frac{a^5 A}{2 x^2}+\frac{1}{8} b^4 x^8 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^3,x]

[Out]

-(a^5*A)/(2*x^2) + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^6)
/6 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^10)/10 + (5*a^4*A*b + a^5*B)*Log[x]

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Maple [A]  time = 0.005, size = 123, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{10}}{10}}+{\frac{A{x}^{8}{b}^{5}}{8}}+{\frac{5\,B{x}^{8}a{b}^{4}}{8}}+{\frac{5\,A{x}^{6}a{b}^{4}}{6}}+{\frac{5\,B{x}^{6}{a}^{2}{b}^{3}}{3}}+{\frac{5\,A{x}^{4}{a}^{2}{b}^{3}}{2}}+{\frac{5\,B{x}^{4}{a}^{3}{b}^{2}}{2}}+5\,A{x}^{2}{a}^{3}{b}^{2}+{\frac{5\,B{x}^{2}{a}^{4}b}{2}}+5\,A\ln \left ( x \right ){a}^{4}b+B\ln \left ( x \right ){a}^{5}-{\frac{A{a}^{5}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^3,x)

[Out]

1/10*b^5*B*x^10+1/8*A*x^8*b^5+5/8*B*x^8*a*b^4+5/6*A*x^6*a*b^4+5/3*B*x^6*a^2*b^3+5/2*A*x^4*a^2*b^3+5/2*B*x^4*a^
3*b^2+5*A*x^2*a^3*b^2+5/2*B*x^2*a^4*b+5*A*ln(x)*a^4*b+B*ln(x)*a^5-1/2*a^5*A/x^2

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Maxima [A]  time = 1.00614, size = 162, normalized size = 1.43 \begin{align*} \frac{1}{10} \, B b^{5} x^{10} + \frac{1}{8} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac{5}{6} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + \frac{5}{2} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} - \frac{A a^{5}}{2 \, x^{2}} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^3,x, algorithm="maxima")

[Out]

1/10*B*b^5*x^10 + 1/8*(5*B*a*b^4 + A*b^5)*x^8 + 5/6*(2*B*a^2*b^3 + A*a*b^4)*x^6 + 5/2*(B*a^3*b^2 + A*a^2*b^3)*
x^4 - 1/2*A*a^5/x^2 + 5/2*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 1/2*(B*a^5 + 5*A*a^4*b)*log(x^2)

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Fricas [A]  time = 1.42959, size = 279, normalized size = 2.47 \begin{align*} \frac{12 \, B b^{5} x^{12} + 15 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 60 \, A a^{5} + 300 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 120 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \log \left (x\right )}{120 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^3,x, algorithm="fricas")

[Out]

1/120*(12*B*b^5*x^12 + 15*(5*B*a*b^4 + A*b^5)*x^10 + 100*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 300*(B*a^3*b^2 + A*a^2*
b^3)*x^6 - 60*A*a^5 + 300*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 120*(B*a^5 + 5*A*a^4*b)*x^2*log(x))/x^2

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Sympy [A]  time = 0.463312, size = 131, normalized size = 1.16 \begin{align*} - \frac{A a^{5}}{2 x^{2}} + \frac{B b^{5} x^{10}}{10} + a^{4} \left (5 A b + B a\right ) \log{\left (x \right )} + x^{8} \left (\frac{A b^{5}}{8} + \frac{5 B a b^{4}}{8}\right ) + x^{6} \left (\frac{5 A a b^{4}}{6} + \frac{5 B a^{2} b^{3}}{3}\right ) + x^{4} \left (\frac{5 A a^{2} b^{3}}{2} + \frac{5 B a^{3} b^{2}}{2}\right ) + x^{2} \left (5 A a^{3} b^{2} + \frac{5 B a^{4} b}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**3,x)

[Out]

-A*a**5/(2*x**2) + B*b**5*x**10/10 + a**4*(5*A*b + B*a)*log(x) + x**8*(A*b**5/8 + 5*B*a*b**4/8) + x**6*(5*A*a*
b**4/6 + 5*B*a**2*b**3/3) + x**4*(5*A*a**2*b**3/2 + 5*B*a**3*b**2/2) + x**2*(5*A*a**3*b**2 + 5*B*a**4*b/2)

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Giac [A]  time = 1.62579, size = 196, normalized size = 1.73 \begin{align*} \frac{1}{10} \, B b^{5} x^{10} + \frac{5}{8} \, B a b^{4} x^{8} + \frac{1}{8} \, A b^{5} x^{8} + \frac{5}{3} \, B a^{2} b^{3} x^{6} + \frac{5}{6} \, A a b^{4} x^{6} + \frac{5}{2} \, B a^{3} b^{2} x^{4} + \frac{5}{2} \, A a^{2} b^{3} x^{4} + \frac{5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left (x^{2}\right ) - \frac{B a^{5} x^{2} + 5 \, A a^{4} b x^{2} + A a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^3,x, algorithm="giac")

[Out]

1/10*B*b^5*x^10 + 5/8*B*a*b^4*x^8 + 1/8*A*b^5*x^8 + 5/3*B*a^2*b^3*x^6 + 5/6*A*a*b^4*x^6 + 5/2*B*a^3*b^2*x^4 +
5/2*A*a^2*b^3*x^4 + 5/2*B*a^4*b*x^2 + 5*A*a^3*b^2*x^2 + 1/2*(B*a^5 + 5*A*a^4*b)*log(x^2) - 1/2*(B*a^5*x^2 + 5*
A*a^4*b*x^2 + A*a^5)/x^2