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

Optimal. Leaf size=125 \[ -\frac {16 c^2 \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^4 x}+\frac {8 c \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^3 x^2}-\frac {2 \sqrt {b x+c x^2} (7 b B-6 A c)}{35 b^2 x^3}-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4} \]

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Rubi [A]  time = 0.11, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 658, 650} \begin {gather*} -\frac {16 c^2 \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^4 x}+\frac {8 c \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^3 x^2}-\frac {2 \sqrt {b x+c x^2} (7 b B-6 A c)}{35 b^2 x^3}-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(7*b*x^4) - (2*(7*b*B - 6*A*c)*Sqrt[b*x + c*x^2])/(35*b^2*x^3) + (8*c*(7*b*B - 6*A*c)
*Sqrt[b*x + c*x^2])/(105*b^3*x^2) - (16*c^2*(7*b*B - 6*A*c)*Sqrt[b*x + c*x^2])/(105*b^4*x)

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]

Rubi steps

\begin {align*} \int \frac {A+B x}{x^4 \sqrt {b x+c x^2}} \, dx &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}+\frac {\left (2 \left (-4 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{7 b}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}-\frac {2 (7 b B-6 A c) \sqrt {b x+c x^2}}{35 b^2 x^3}-\frac {(4 c (7 b B-6 A c)) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{35 b^2}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}-\frac {2 (7 b B-6 A c) \sqrt {b x+c x^2}}{35 b^2 x^3}+\frac {8 c (7 b B-6 A c) \sqrt {b x+c x^2}}{105 b^3 x^2}+\frac {\left (8 c^2 (7 b B-6 A c)\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}-\frac {2 (7 b B-6 A c) \sqrt {b x+c x^2}}{35 b^2 x^3}+\frac {8 c (7 b B-6 A c) \sqrt {b x+c x^2}}{105 b^3 x^2}-\frac {16 c^2 (7 b B-6 A c) \sqrt {b x+c x^2}}{105 b^4 x}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 79, normalized size = 0.63 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (3 A \left (5 b^3-6 b^2 c x+8 b c^2 x^2-16 c^3 x^3\right )+7 b B x \left (3 b^2-4 b c x+8 c^2 x^2\right )\right )}{105 b^4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.35, size = 84, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-15 A b^3+18 A b^2 c x-24 A b c^2 x^2+48 A c^3 x^3-21 b^3 B x+28 b^2 B c x^2-56 b B c^2 x^3\right )}{105 b^4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 82, normalized size = 0.66 \begin {gather*} -\frac {2 \, {\left (15 \, A b^{3} + 8 \, {\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} - 4 \, {\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, b^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 191, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b \sqrt {c} + 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{\frac {3}{2}} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} + 252 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} \sqrt {c} + 15 \, A b^{3}\right )}}{105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 86, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+56 B b \,c^{2} x^{3}+24 A b \,c^{2} x^{2}-28 B \,b^{2} c \,x^{2}-18 A \,b^{2} c x +21 B \,b^{3} x +15 A \,b^{3}\right )}{105 \sqrt {c \,x^{2}+b x}\, b^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.93, size = 152, normalized size = 1.22 \begin {gather*} -\frac {16 \, \sqrt {c x^{2} + b x} B c^{2}}{15 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{3}}{35 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c}{15 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{2}}{35 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{5 \, b x^{3}} + \frac {12 \, \sqrt {c x^{2} + b x} A c}{35 \, b^{2} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{7 \, b x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/15*sqrt(c*x^2 + b*x)*B*c^2/(b^3*x) + 32/35*sqrt(c*x^2 + b*x)*A*c^3/(b^4*x) + 8/15*sqrt(c*x^2 + b*x)*B*c/(b
^2*x^2) - 16/35*sqrt(c*x^2 + b*x)*A*c^2/(b^3*x^2) - 2/5*sqrt(c*x^2 + b*x)*B/(b*x^3) + 12/35*sqrt(c*x^2 + b*x)*
A*c/(b^2*x^3) - 2/7*sqrt(c*x^2 + b*x)*A/(b*x^4)

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mupad [B]  time = 1.12, size = 113, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c\,x^2+b\,x}\,\left (96\,A\,c^3-112\,B\,b\,c^2\right )}{105\,b^4\,x}-\frac {\left (48\,A\,c^2-56\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x^2}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{7\,b\,x^4}+\frac {\sqrt {c\,x^2+b\,x}\,\left (12\,A\,c-14\,B\,b\right )}{35\,b^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + c*x^2)^(1/2)*(96*A*c^3 - 112*B*b*c^2))/(105*b^4*x) - ((48*A*c^2 - 56*B*b*c)*(b*x + c*x^2)^(1/2))/(105*
b^3*x^2) - (2*A*(b*x + c*x^2)^(1/2))/(7*b*x^4) + ((b*x + c*x^2)^(1/2)*(12*A*c - 14*B*b))/(35*b^2*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{4} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)/(x**4*sqrt(x*(b + c*x))), x)

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