3.1.47 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{15}} \, dx\) [47]

Optimal. Leaf size=48 \[ -\frac {A \left (a+b x^2\right )^6}{14 a x^{14}}+\frac {(A b-7 a B) \left (a+b x^2\right )^6}{84 a^2 x^{12}} \]

[Out]

-1/14*A*(b*x^2+a)^6/a/x^14+1/84*(A*b-7*B*a)*(b*x^2+a)^6/a^2/x^12

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Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {457, 79, 37} \begin {gather*} \frac {\left (a+b x^2\right )^6 (A b-7 a B)}{84 a^2 x^{12}}-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^15,x]

[Out]

-1/14*(A*(a + b*x^2)^6)/(a*x^14) + ((A*b - 7*a*B)*(a + b*x^2)^6)/(84*a^2*x^12)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

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

Rule 457

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 {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{15}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}}+\frac {(-A b+7 a B) \text {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^2\right )}{14 a}\\ &=-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}}+\frac {(A b-7 a B) \left (a+b x^2\right )^6}{84 a^2 x^{12}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(118\) vs. \(2(48)=96\).
time = 0.02, size = 118, normalized size = 2.46 \begin {gather*} -\frac {21 b^5 x^{10} \left (A+2 B x^2\right )+35 a b^4 x^8 \left (2 A+3 B x^2\right )+35 a^2 b^3 x^6 \left (3 A+4 B x^2\right )+21 a^3 b^2 x^4 \left (4 A+5 B x^2\right )+7 a^4 b x^2 \left (5 A+6 B x^2\right )+a^5 \left (6 A+7 B x^2\right )}{84 x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^15,x]

[Out]

-1/84*(21*b^5*x^10*(A + 2*B*x^2) + 35*a*b^4*x^8*(2*A + 3*B*x^2) + 35*a^2*b^3*x^6*(3*A + 4*B*x^2) + 21*a^3*b^2*
x^4*(4*A + 5*B*x^2) + 7*a^4*b*x^2*(5*A + 6*B*x^2) + a^5*(6*A + 7*B*x^2))/x^14

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(44)=88\).
time = 0.07, size = 104, normalized size = 2.17

method	result	size
default	\(-\frac {a^{5} A}{14 x^{14}}-\frac {b^{4} \left (A b +5 B a \right )}{4 x^{4}}-\frac {a^{4} \left (5 A b +B a \right )}{12 x^{12}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{6 x^{6}}-\frac {b^{5} B}{2 x^{2}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{4 x^{8}}-\frac {a^{3} b \left (2 A b +B a \right )}{2 x^{10}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{14}+\left (-\frac {5}{12} a^{4} b A -\frac {1}{12} a^{5} B \right ) x^{2}+\left (-a^{3} b^{2} A -\frac {1}{2} a^{4} b B \right ) x^{4}+\left (-\frac {5}{4} a^{2} b^{3} A -\frac {5}{4} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{6} a \,b^{4} A -\frac {5}{3} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{4} b^{5} A -\frac {5}{4} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{2}}{x^{14}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{14}+\left (-\frac {5}{12} a^{4} b A -\frac {1}{12} a^{5} B \right ) x^{2}+\left (-a^{3} b^{2} A -\frac {1}{2} a^{4} b B \right ) x^{4}+\left (-\frac {5}{4} a^{2} b^{3} A -\frac {5}{4} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{6} a \,b^{4} A -\frac {5}{3} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{4} b^{5} A -\frac {5}{4} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{2}}{x^{14}}\)	\(122\)
gosper	\(-\frac {42 b^{5} B \,x^{12}+21 A \,b^{5} x^{10}+105 B a \,b^{4} x^{10}+70 a A \,b^{4} x^{8}+140 B \,a^{2} b^{3} x^{8}+105 a^{2} A \,b^{3} x^{6}+105 B \,a^{3} b^{2} x^{6}+84 a^{3} A \,b^{2} x^{4}+42 B \,a^{4} b \,x^{4}+35 a^{4} A b \,x^{2}+7 B \,a^{5} x^{2}+6 a^{5} A}{84 x^{14}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^15,x,method=_RETURNVERBOSE)

[Out]

-1/14*a^5*A/x^14-1/4*b^4*(A*b+5*B*a)/x^4-1/12*a^4*(5*A*b+B*a)/x^12-5/6*a*b^3*(A*b+2*B*a)/x^6-1/2*b^5*B/x^2-5/4
*a^2*b^2*(A*b+B*a)/x^8-1/2*a^3*b*(2*A*b+B*a)/x^10

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (45) = 90\).
time = 0.32, size = 121, normalized size = 2.52 \begin {gather*} -\frac {42 \, B b^{5} x^{12} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 6 \, A a^{5} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^15,x, algorithm="maxima")

[Out]

-1/84*(42*B*b^5*x^12 + 21*(5*B*a*b^4 + A*b^5)*x^10 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 105*(B*a^3*b^2 + A*a^2*b
^3)*x^6 + 6*A*a^5 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 7*(B*a^5 + 5*A*a^4*b)*x^2)/x^14

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (45) = 90\).
time = 0.77, size = 121, normalized size = 2.52 \begin {gather*} -\frac {42 \, B b^{5} x^{12} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 6 \, A a^{5} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^15,x, algorithm="fricas")

[Out]

-1/84*(42*B*b^5*x^12 + 21*(5*B*a*b^4 + A*b^5)*x^10 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 105*(B*a^3*b^2 + A*a^2*b
^3)*x^6 + 6*A*a^5 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 7*(B*a^5 + 5*A*a^4*b)*x^2)/x^14

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (41) = 82\).
time = 134.48, size = 134, normalized size = 2.79 \begin {gather*} \frac {- 6 A a^{5} - 42 B b^{5} x^{12} + x^{10} \left (- 21 A b^{5} - 105 B a b^{4}\right ) + x^{8} \left (- 70 A a b^{4} - 140 B a^{2} b^{3}\right ) + x^{6} \left (- 105 A a^{2} b^{3} - 105 B a^{3} b^{2}\right ) + x^{4} \left (- 84 A a^{3} b^{2} - 42 B a^{4} b\right ) + x^{2} \left (- 35 A a^{4} b - 7 B a^{5}\right )}{84 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**15,x)

[Out]

(-6*A*a**5 - 42*B*b**5*x**12 + x**10*(-21*A*b**5 - 105*B*a*b**4) + x**8*(-70*A*a*b**4 - 140*B*a**2*b**3) + x**
6*(-105*A*a**2*b**3 - 105*B*a**3*b**2) + x**4*(-84*A*a**3*b**2 - 42*B*a**4*b) + x**2*(-35*A*a**4*b - 7*B*a**5)
)/(84*x**14)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (45) = 90\).
time = 1.10, size = 127, normalized size = 2.65 \begin {gather*} -\frac {42 \, B b^{5} x^{12} + 105 \, B a b^{4} x^{10} + 21 \, A b^{5} x^{10} + 140 \, B a^{2} b^{3} x^{8} + 70 \, A a b^{4} x^{8} + 105 \, B a^{3} b^{2} x^{6} + 105 \, A a^{2} b^{3} x^{6} + 42 \, B a^{4} b x^{4} + 84 \, A a^{3} b^{2} x^{4} + 7 \, B a^{5} x^{2} + 35 \, A a^{4} b x^{2} + 6 \, A a^{5}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^15,x, algorithm="giac")

[Out]

-1/84*(42*B*b^5*x^12 + 105*B*a*b^4*x^10 + 21*A*b^5*x^10 + 140*B*a^2*b^3*x^8 + 70*A*a*b^4*x^8 + 105*B*a^3*b^2*x
^6 + 105*A*a^2*b^3*x^6 + 42*B*a^4*b*x^4 + 84*A*a^3*b^2*x^4 + 7*B*a^5*x^2 + 35*A*a^4*b*x^2 + 6*A*a^5)/x^14

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Mupad [B]
time = 0.03, size = 121, normalized size = 2.52 \begin {gather*} -\frac {\frac {A\,a^5}{14}+x^4\,\left (\frac {B\,a^4\,b}{2}+A\,a^3\,b^2\right )+x^8\,\left (\frac {5\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{6}\right )+x^2\,\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )+x^{10}\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{4}+\frac {5\,A\,a^2\,b^3}{4}\right )+\frac {B\,b^5\,x^{12}}{2}}{x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^15,x)

[Out]

-((A*a^5)/14 + x^4*(A*a^3*b^2 + (B*a^4*b)/2) + x^8*((5*B*a^2*b^3)/3 + (5*A*a*b^4)/6) + x^2*((B*a^5)/12 + (5*A*
a^4*b)/12) + x^10*((A*b^5)/4 + (5*B*a*b^4)/4) + x^6*((5*A*a^2*b^3)/4 + (5*B*a^3*b^2)/4) + (B*b^5*x^12)/2)/x^14

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