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

Optimal. Leaf size=112 \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+a^4 x (a B+5 A b)-\frac{a^5 A}{2 x^2}+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

[Out]

-(a^5*A)/(2*x^2) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (a*b
^3*(A*b + 2*a*B)*x^10)/2 + (b^4*(A*b + 5*a*B)*x^13)/13 + (b^5*B*x^16)/16

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Rubi [A]  time = 0.0606458, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+a^4 x (a B+5 A b)-\frac{a^5 A}{2 x^2}+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*A)/(2*x^2) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (a*b
^3*(A*b + 2*a*B)*x^10)/2 + (b^4*(A*b + 5*a*B)*x^13)/13 + (b^5*B*x^16)/16

Rule 448

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^3} \, dx &=\int \left (a^4 (5 A b+a B)+\frac{a^5 A}{x^3}+5 a^3 b (2 A b+a B) x^3+10 a^2 b^2 (A b+a B) x^6+5 a b^3 (A b+2 a B) x^9+b^4 (A b+5 a B) x^{12}+b^5 B x^{15}\right ) \, dx\\ &=-\frac{a^5 A}{2 x^2}+a^4 (5 A b+a B) x+\frac{5}{4} a^3 b (2 A b+a B) x^4+\frac{10}{7} a^2 b^2 (A b+a B) x^7+\frac{1}{2} a b^3 (A b+2 a B) x^{10}+\frac{1}{13} b^4 (A b+5 a B) x^{13}+\frac{1}{16} b^5 B x^{16}\\ \end{align*}

Mathematica [A]  time = 0.0301264, size = 112, normalized size = 1. \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+a^4 x (a B+5 A b)-\frac{a^5 A}{2 x^2}+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*A)/(2*x^2) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (a*b
^3*(A*b + 2*a*B)*x^10)/2 + (b^4*(A*b + 5*a*B)*x^13)/13 + (b^5*B*x^16)/16

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Maple [A]  time = 0.004, size = 120, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{16}}{16}}+{\frac{A{x}^{13}{b}^{5}}{13}}+{\frac{5\,B{x}^{13}a{b}^{4}}{13}}+{\frac{A{x}^{10}a{b}^{4}}{2}}+B{x}^{10}{a}^{2}{b}^{3}+{\frac{10\,A{x}^{7}{a}^{2}{b}^{3}}{7}}+{\frac{10\,B{x}^{7}{a}^{3}{b}^{2}}{7}}+{\frac{5\,A{x}^{4}{a}^{3}{b}^{2}}{2}}+{\frac{5\,B{x}^{4}{a}^{4}b}{4}}+5\,{a}^{4}bAx+{a}^{5}Bx-{\frac{A{a}^{5}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^5*B*x^16+1/13*A*x^13*b^5+5/13*B*x^13*a*b^4+1/2*A*x^10*a*b^4+B*x^10*a^2*b^3+10/7*A*x^7*a^2*b^3+10/7*B*x^
7*a^3*b^2+5/2*A*x^4*a^3*b^2+5/4*B*x^4*a^4*b+5*a^4*b*A*x+a^5*B*x-1/2*a^5*A/x^2

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Maxima [A]  time = 1.11409, size = 157, normalized size = 1.4 \begin{align*} \frac{1}{16} \, B b^{5} x^{16} + \frac{1}{13} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{13} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{10}{7} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + \frac{5}{4} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - \frac{A a^{5}}{2 \, x^{2}} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*B*b^5*x^16 + 1/13*(5*B*a*b^4 + A*b^5)*x^13 + 1/2*(2*B*a^2*b^3 + A*a*b^4)*x^10 + 10/7*(B*a^3*b^2 + A*a^2*b
^3)*x^7 + 5/4*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 1/2*A*a^5/x^2 + (B*a^5 + 5*A*a^4*b)*x

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Fricas [A]  time = 1.3694, size = 279, normalized size = 2.49 \begin{align*} \frac{91 \, B b^{5} x^{18} + 112 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 728 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 1820 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 728 \, A a^{5} + 1456 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1456*(91*B*b^5*x^18 + 112*(5*B*a*b^4 + A*b^5)*x^15 + 728*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 2080*(B*a^3*b^2 + A*
a^2*b^3)*x^9 + 1820*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 728*A*a^5 + 1456*(B*a^5 + 5*A*a^4*b)*x^3)/x^2

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Sympy [A]  time = 0.377499, size = 128, normalized size = 1.14 \begin{align*} - \frac{A a^{5}}{2 x^{2}} + \frac{B b^{5} x^{16}}{16} + x^{13} \left (\frac{A b^{5}}{13} + \frac{5 B a b^{4}}{13}\right ) + x^{10} \left (\frac{A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{7} \left (\frac{10 A a^{2} b^{3}}{7} + \frac{10 B a^{3} b^{2}}{7}\right ) + x^{4} \left (\frac{5 A a^{3} b^{2}}{2} + \frac{5 B a^{4} b}{4}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a**5/(2*x**2) + B*b**5*x**16/16 + x**13*(A*b**5/13 + 5*B*a*b**4/13) + x**10*(A*a*b**4/2 + B*a**2*b**3) + x*
*7*(10*A*a**2*b**3/7 + 10*B*a**3*b**2/7) + x**4*(5*A*a**3*b**2/2 + 5*B*a**4*b/4) + x*(5*A*a**4*b + B*a**5)

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Giac [A]  time = 1.22865, size = 161, normalized size = 1.44 \begin{align*} \frac{1}{16} \, B b^{5} x^{16} + \frac{5}{13} \, B a b^{4} x^{13} + \frac{1}{13} \, A b^{5} x^{13} + B a^{2} b^{3} x^{10} + \frac{1}{2} \, A a b^{4} x^{10} + \frac{10}{7} \, B a^{3} b^{2} x^{7} + \frac{10}{7} \, A a^{2} b^{3} x^{7} + \frac{5}{4} \, B a^{4} b x^{4} + \frac{5}{2} \, A a^{3} b^{2} x^{4} + B a^{5} x + 5 \, A a^{4} b x - \frac{A a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*B*b^5*x^16 + 5/13*B*a*b^4*x^13 + 1/13*A*b^5*x^13 + B*a^2*b^3*x^10 + 1/2*A*a*b^4*x^10 + 10/7*B*a^3*b^2*x^7
+ 10/7*A*a^2*b^3*x^7 + 5/4*B*a^4*b*x^4 + 5/2*A*a^3*b^2*x^4 + B*a^5*x + 5*A*a^4*b*x - 1/2*A*a^5/x^2