3.2.37 \(\int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx\)

Optimal. Leaf size=133 \[ -\frac {8 c^2 \sqrt {b x^2+c x^4} (7 b B-6 A c)}{105 b^4 x^2}+\frac {4 c \sqrt {b x^2+c x^4} (7 b B-6 A c)}{105 b^3 x^4}-\frac {\sqrt {b x^2+c x^4} (7 b B-6 A c)}{35 b^2 x^6}-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8} \]

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Rubi [A]  time = 0.25, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \begin {gather*} -\frac {8 c^2 \sqrt {b x^2+c x^4} (7 b B-6 A c)}{105 b^4 x^2}+\frac {4 c \sqrt {b x^2+c x^4} (7 b B-6 A c)}{105 b^3 x^4}-\frac {\sqrt {b x^2+c x^4} (7 b B-6 A c)}{35 b^2 x^6}-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^7*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-(A*Sqrt[b*x^2 + c*x^4])/(7*b*x^8) - ((7*b*B - 6*A*c)*Sqrt[b*x^2 + c*x^4])/(35*b^2*x^6) + (4*c*(7*b*B - 6*A*c)
*Sqrt[b*x^2 + c*x^4])/(105*b^3*x^4) - (8*c^2*(7*b*B - 6*A*c)*Sqrt[b*x^2 + c*x^4])/(105*b^4*x^2)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 2034

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n
, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x]
/; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && Integ
erQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^4 \sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8}+\frac {\left (-4 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx,x,x^2\right )}{7 b}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8}-\frac {(7 b B-6 A c) \sqrt {b x^2+c x^4}}{35 b^2 x^6}-\frac {(2 c (7 b B-6 A c)) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx,x,x^2\right )}{35 b^2}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8}-\frac {(7 b B-6 A c) \sqrt {b x^2+c x^4}}{35 b^2 x^6}+\frac {4 c (7 b B-6 A c) \sqrt {b x^2+c x^4}}{105 b^3 x^4}+\frac {\left (4 c^2 (7 b B-6 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+c x^2}} \, dx,x,x^2\right )}{105 b^3}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8}-\frac {(7 b B-6 A c) \sqrt {b x^2+c x^4}}{35 b^2 x^6}+\frac {4 c (7 b B-6 A c) \sqrt {b x^2+c x^4}}{105 b^3 x^4}-\frac {8 c^2 (7 b B-6 A c) \sqrt {b x^2+c x^4}}{105 b^4 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 89, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (3 A \left (5 b^3-6 b^2 c x^2+8 b c^2 x^4-16 c^3 x^6\right )+7 b B x^2 \left (3 b^2-4 b c x^2+8 c^2 x^4\right )\right )}{105 b^4 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^7*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-1/105*(Sqrt[x^2*(b + c*x^2)]*(7*b*B*x^2*(3*b^2 - 4*b*c*x^2 + 8*c^2*x^4) + 3*A*(5*b^3 - 6*b^2*c*x^2 + 8*b*c^2*
x^4 - 16*c^3*x^6)))/(b^4*x^8)

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IntegrateAlgebraic [A]  time = 0.37, size = 90, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-15 A b^3+18 A b^2 c x^2-24 A b c^2 x^4+48 A c^3 x^6-21 b^3 B x^2+28 b^2 B c x^4-56 b B c^2 x^6\right )}{105 b^4 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x^2)/(x^7*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(Sqrt[b*x^2 + c*x^4]*(-15*A*b^3 - 21*b^3*B*x^2 + 18*A*b^2*c*x^2 + 28*b^2*B*c*x^4 - 24*A*b*c^2*x^4 - 56*b*B*c^2
*x^6 + 48*A*c^3*x^6))/(105*b^4*x^8)

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fricas [A]  time = 0.42, size = 86, normalized size = 0.65 \begin {gather*} -\frac {{\left (8 \, {\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{6} - 4 \, {\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{4} + 15 \, A b^{3} + 3 \, {\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{105 \, b^{4} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(8*(7*B*b*c^2 - 6*A*c^3)*x^6 - 4*(7*B*b^2*c - 6*A*b*c^2)*x^4 + 15*A*b^3 + 3*(7*B*b^3 - 6*A*b^2*c)*x^2)*
sqrt(c*x^4 + b*x^2)/(b^4*x^8)

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giac [A]  time = 0.31, size = 219, normalized size = 1.65 \begin {gather*} \frac {140 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{4} B c + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{3} B b \sqrt {c} + 210 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{3} A c^{\frac {3}{2}} + 21 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{2} B b^{2} + 252 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{2} A b c + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} A b^{2} \sqrt {c} + 15 \, A b^{3}}{105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

1/105*(140*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))^4*B*c + 105*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))^3*B*b*sqrt(c) +
210*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))^3*A*c^(3/2) + 21*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))^2*B*b^2 + 252*(s
qrt(c)*x^2 - sqrt(c*x^4 + b*x^2))^2*A*b*c + 105*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))*A*b^2*sqrt(c) + 15*A*b^3)/
(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))^7

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maple [A]  time = 0.05, size = 94, normalized size = 0.71 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-48 A \,c^{3} x^{6}+56 B b \,c^{2} x^{6}+24 A b \,c^{2} x^{4}-28 B \,b^{2} c \,x^{4}-18 A \,b^{2} c \,x^{2}+21 B \,b^{3} x^{2}+15 A \,b^{3}\right )}{105 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/105*(c*x^2+b)*(-48*A*c^3*x^6+56*B*b*c^2*x^6+24*A*b*c^2*x^4-28*B*b^2*c*x^4-18*A*b^2*c*x^2+21*B*b^3*x^2+15*A*
b^3)/x^6/b^4/(c*x^4+b*x^2)^(1/2)

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maxima [A]  time = 1.49, size = 167, normalized size = 1.26 \begin {gather*} -\frac {1}{15} \, B {\left (\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{3} x^{2}} - \frac {4 \, \sqrt {c x^{4} + b x^{2}} c}{b^{2} x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}}}{b x^{6}}\right )} + \frac {1}{35} \, A {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{4} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{3} x^{4}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} c}{b^{2} x^{6}} - \frac {5 \, \sqrt {c x^{4} + b x^{2}}}{b x^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

-1/15*B*(8*sqrt(c*x^4 + b*x^2)*c^2/(b^3*x^2) - 4*sqrt(c*x^4 + b*x^2)*c/(b^2*x^4) + 3*sqrt(c*x^4 + b*x^2)/(b*x^
6)) + 1/35*A*(16*sqrt(c*x^4 + b*x^2)*c^3/(b^4*x^2) - 8*sqrt(c*x^4 + b*x^2)*c^2/(b^3*x^4) + 6*sqrt(c*x^4 + b*x^
2)*c/(b^2*x^6) - 5*sqrt(c*x^4 + b*x^2)/(b*x^8))

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mupad [B]  time = 0.30, size = 121, normalized size = 0.91 \begin {gather*} \frac {\left (6\,A\,c-7\,B\,b\right )\,\sqrt {c\,x^4+b\,x^2}}{35\,b^2\,x^6}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{7\,b\,x^8}-\frac {\left (24\,A\,c^2-28\,B\,b\,c\right )\,\sqrt {c\,x^4+b\,x^2}}{105\,b^3\,x^4}+\frac {\left (48\,A\,c^3-56\,B\,b\,c^2\right )\,\sqrt {c\,x^4+b\,x^2}}{105\,b^4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^7*(b*x^2 + c*x^4)^(1/2)),x)

[Out]

((6*A*c - 7*B*b)*(b*x^2 + c*x^4)^(1/2))/(35*b^2*x^6) - (A*(b*x^2 + c*x^4)^(1/2))/(7*b*x^8) - ((24*A*c^2 - 28*B
*b*c)*(b*x^2 + c*x^4)^(1/2))/(105*b^3*x^4) + ((48*A*c^3 - 56*B*b*c^2)*(b*x^2 + c*x^4)^(1/2))/(105*b^4*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**7/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral((A + B*x**2)/(x**7*sqrt(x**2*(b + c*x**2))), x)

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