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

Optimal. Leaf size=88 \[ \frac{5}{3} a^2 A b^3 x^6+\frac{5}{2} a^3 A b^2 x^4+\frac{5}{2} a^4 A b x^2+a^5 A \log (x)+\frac{5}{8} a A b^4 x^8+\frac{B \left (a+b x^2\right )^6}{12 b}+\frac{1}{10} A b^5 x^{10} \]

[Out]

(5*a^4*A*b*x^2)/2 + (5*a^3*A*b^2*x^4)/2 + (5*a^2*A*b^3*x^6)/3 + (5*a*A*b^4*x^8)/8 + (A*b^5*x^10)/10 + (B*(a +
b*x^2)^6)/(12*b) + a^5*A*Log[x]

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Rubi [A]  time = 0.0645913, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 80, 43} \[ \frac{5}{3} a^2 A b^3 x^6+\frac{5}{2} a^3 A b^2 x^4+\frac{5}{2} a^4 A b x^2+a^5 A \log (x)+\frac{5}{8} a A b^4 x^8+\frac{B \left (a+b x^2\right )^6}{12 b}+\frac{1}{10} A b^5 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x,x]

[Out]

(5*a^4*A*b*x^2)/2 + (5*a^3*A*b^2*x^4)/2 + (5*a^2*A*b^3*x^6)/3 + (5*a*A*b^4*x^8)/8 + (A*b^5*x^10)/10 + (B*(a +
b*x^2)^6)/(12*b) + a^5*A*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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x} \, dx,x,x^2\right )\\ &=\frac{B \left (a+b x^2\right )^6}{12 b}+\frac{1}{2} A \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x} \, dx,x,x^2\right )\\ &=\frac{B \left (a+b x^2\right )^6}{12 b}+\frac{1}{2} A \operatorname{Subst}\left (\int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^2\right )\\ &=\frac{5}{2} a^4 A b x^2+\frac{5}{2} a^3 A b^2 x^4+\frac{5}{3} a^2 A b^3 x^6+\frac{5}{8} a A b^4 x^8+\frac{1}{10} A b^5 x^{10}+\frac{B \left (a+b x^2\right )^6}{12 b}+a^5 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.028904, size = 113, normalized size = 1.28 \[ \frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^5 A \log (x)+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{12} b^5 B x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(5*A*b + a*B)*x^2)/2 + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (5*a^2*b^2*(A*b + a*B)*x^6)/3 + (5*a*b^3*(A*b + 2*
a*B)*x^8)/8 + (b^4*(A*b + 5*a*B)*x^10)/10 + (b^5*B*x^12)/12 + a^5*A*Log[x]

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Maple [A]  time = 0.003, size = 124, normalized size = 1.4 \begin{align*}{\frac{B{b}^{5}{x}^{12}}{12}}+{\frac{A{b}^{5}{x}^{10}}{10}}+{\frac{B{x}^{10}a{b}^{4}}{2}}+{\frac{5\,aA{b}^{4}{x}^{8}}{8}}+{\frac{5\,B{x}^{8}{a}^{2}{b}^{3}}{4}}+{\frac{5\,{a}^{2}A{b}^{3}{x}^{6}}{3}}+{\frac{5\,B{x}^{6}{a}^{3}{b}^{2}}{3}}+{\frac{5\,{a}^{3}A{b}^{2}{x}^{4}}{2}}+{\frac{5\,B{x}^{4}{a}^{4}b}{4}}+{\frac{5\,{a}^{4}Ab{x}^{2}}{2}}+{\frac{B{x}^{2}{a}^{5}}{2}}+{a}^{5}A\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*B*b^5*x^12+1/10*A*b^5*x^10+1/2*B*x^10*a*b^4+5/8*a*A*b^4*x^8+5/4*B*x^8*a^2*b^3+5/3*a^2*A*b^3*x^6+5/3*B*x^6
*a^3*b^2+5/2*a^3*A*b^2*x^4+5/4*B*x^4*a^4*b+5/2*a^4*A*b*x^2+1/2*B*x^2*a^5+a^5*A*ln(x)

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Maxima [A]  time = 1.02679, size = 162, normalized size = 1.84 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{10} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac{5}{8} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac{5}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + \frac{1}{2} \, A a^{5} \log \left (x^{2}\right ) + \frac{5}{4} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} 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,x, algorithm="maxima")

[Out]

1/12*B*b^5*x^12 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/8*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 5/3*(B*a^3*b^2 + A*a^2*b^3
)*x^6 + 1/2*A*a^5*log(x^2) + 5/4*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1/2*(B*a^5 + 5*A*a^4*b)*x^2

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Fricas [A]  time = 1.45067, size = 265, normalized size = 3.01 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{10} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac{5}{8} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac{5}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + A a^{5} \log \left (x\right ) + \frac{5}{4} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} 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,x, algorithm="fricas")

[Out]

1/12*B*b^5*x^12 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/8*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 5/3*(B*a^3*b^2 + A*a^2*b^3
)*x^6 + A*a^5*log(x) + 5/4*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1/2*(B*a^5 + 5*A*a^4*b)*x^2

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Sympy [A]  time = 0.368107, size = 134, normalized size = 1.52 \begin{align*} A a^{5} \log{\left (x \right )} + \frac{B b^{5} x^{12}}{12} + x^{10} \left (\frac{A b^{5}}{10} + \frac{B a b^{4}}{2}\right ) + x^{8} \left (\frac{5 A a b^{4}}{8} + \frac{5 B a^{2} b^{3}}{4}\right ) + x^{6} \left (\frac{5 A a^{2} b^{3}}{3} + \frac{5 B a^{3} b^{2}}{3}\right ) + x^{4} \left (\frac{5 A a^{3} b^{2}}{2} + \frac{5 B a^{4} b}{4}\right ) + x^{2} \left (\frac{5 A a^{4} b}{2} + \frac{B a^{5}}{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,x)

[Out]

A*a**5*log(x) + B*b**5*x**12/12 + x**10*(A*b**5/10 + B*a*b**4/2) + x**8*(5*A*a*b**4/8 + 5*B*a**2*b**3/4) + x**
6*(5*A*a**2*b**3/3 + 5*B*a**3*b**2/3) + x**4*(5*A*a**3*b**2/2 + 5*B*a**4*b/4) + x**2*(5*A*a**4*b/2 + B*a**5/2)

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Giac [A]  time = 1.33182, size = 170, normalized size = 1.93 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{2} \, B a b^{4} x^{10} + \frac{1}{10} \, A b^{5} x^{10} + \frac{5}{4} \, B a^{2} b^{3} x^{8} + \frac{5}{8} \, A a b^{4} x^{8} + \frac{5}{3} \, B a^{3} b^{2} x^{6} + \frac{5}{3} \, A a^{2} b^{3} x^{6} + \frac{5}{4} \, B a^{4} b x^{4} + \frac{5}{2} \, A a^{3} b^{2} x^{4} + \frac{1}{2} \, B a^{5} x^{2} + \frac{5}{2} \, A a^{4} b x^{2} + \frac{1}{2} \, A a^{5} \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,x, algorithm="giac")

[Out]

1/12*B*b^5*x^12 + 1/2*B*a*b^4*x^10 + 1/10*A*b^5*x^10 + 5/4*B*a^2*b^3*x^8 + 5/8*A*a*b^4*x^8 + 5/3*B*a^3*b^2*x^6
+ 5/3*A*a^2*b^3*x^6 + 5/4*B*a^4*b*x^4 + 5/2*A*a^3*b^2*x^4 + 1/2*B*a^5*x^2 + 5/2*A*a^4*b*x^2 + 1/2*A*a^5*log(x
^2)