3.6.99 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=200 \[ \frac {3 a b x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {b^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{2 (a+b x)}+\frac {a^2 \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}+\frac {b^3 B x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

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Rubi [A]  time = 0.09, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} \frac {3 a b x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {b^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{2 (a+b x)}+\frac {a^2 \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^3 B x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^2,x]

[Out]

-((a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x))) + (3*a*b*(A*b + a*B)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
a + b*x) + (b^2*(A*b + 3*a*B)*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (b^3*B*x^3*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*(a + b*x)) + (a^2*(3*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{x^2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (3 a b^4 (A b+a B)+\frac {a^3 A b^3}{x^2}+\frac {a^2 b^3 (3 A b+a B)}{x}+b^5 (A b+3 a B) x+b^6 B x^2\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b (A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^2 (A b+3 a B) x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 B x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 89, normalized size = 0.44 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-6 a^3 A+6 a^2 x \log (x) (a B+3 A b)+18 a^2 b B x^2+9 a b^2 x^2 (2 A+B x)+b^3 x^3 (3 A+2 B x)\right )}{6 x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^2,x]

[Out]

(Sqrt[(a + b*x)^2]*(-6*a^3*A + 18*a^2*b*B*x^2 + 9*a*b^2*x^2*(2*A + B*x) + b^3*x^3*(3*A + 2*B*x) + 6*a^2*(3*A*b
+ a*B)*x*Log[x]))/(6*x*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.29, size = 543, normalized size = 2.72 \begin {gather*} -\frac {3}{2} a^2 A \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-\frac {3}{2} a^2 A \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )+3 a^2 A b \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )-\frac {a^3 \sqrt {b^2} B \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 b}-\frac {a^3 \sqrt {b^2} B \log \left (b \sqrt {a^2+2 a b x+b^2 x^2}-a b-\sqrt {b^2} b x\right )}{2 b}+a^3 B \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-24 a^3 A b+28 a^3 b B x-63 a^2 A b^2 x+72 a^2 b^2 B x^2+72 a A b^3 x^2+36 a b^3 B x^3+12 A b^4 x^3+8 b^4 B x^4\right )+\sqrt {b^2} \left (24 a^4 A-28 a^4 B x+87 a^3 A b x-100 a^3 b B x^2-9 a^2 A b^2 x^2-108 a^2 b^2 B x^3-84 a A b^3 x^3-44 a b^3 B x^4-12 A b^4 x^4-8 b^4 B x^5\right )}{24 x \left (a b+b^2 x\right )-24 \sqrt {b^2} x \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^2,x]

[Out]

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-24*a^3*A*b - 63*a^2*A*b^2*x + 28*a^3*b*B*x + 72*a*A*b^3*x^2 + 72*a^2*b^2*B*x^
2 + 12*A*b^4*x^3 + 36*a*b^3*B*x^3 + 8*b^4*B*x^4) + Sqrt[b^2]*(24*a^4*A + 87*a^3*A*b*x - 28*a^4*B*x - 9*a^2*A*b
^2*x^2 - 100*a^3*b*B*x^2 - 84*a*A*b^3*x^3 - 108*a^2*b^2*B*x^3 - 12*A*b^4*x^4 - 44*a*b^3*B*x^4 - 8*b^4*B*x^5))/
(24*x*(a*b + b^2*x) - 24*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 3*a^2*A*b*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt
[a^2 + 2*a*b*x + b^2*x^2]/a] + a^3*B*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a] - (3*a^2*A*Sqr
t[b^2]*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/2 - (3*a^2*A*Sqrt[b^2]*Log[a - Sqrt[b^2]*x + Sqr
t[a^2 + 2*a*b*x + b^2*x^2]])/2 - (a^3*Sqrt[b^2]*B*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/(2*b)
- (a^3*Sqrt[b^2]*B*Log[-(a*b) - b*Sqrt[b^2]*x + b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/(2*b)

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fricas [A]  time = 0.42, size = 75, normalized size = 0.38 \begin {gather*} \frac {2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \relax (x)}{6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^2,x, algorithm="fricas")

[Out]

1/6*(2*B*b^3*x^4 - 6*A*a^3 + 3*(3*B*a*b^2 + A*b^3)*x^3 + 18*(B*a^2*b + A*a*b^2)*x^2 + 6*(B*a^3 + 3*A*a^2*b)*x*
log(x))/x

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giac [A]  time = 0.18, size = 119, normalized size = 0.60 \begin {gather*} \frac {1}{3} \, B b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} x \mathrm {sgn}\left (b x + a\right ) - \frac {A a^{3} \mathrm {sgn}\left (b x + a\right )}{x} + {\left (B a^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{2} b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^2,x, algorithm="giac")

[Out]

1/3*B*b^3*x^3*sgn(b*x + a) + 3/2*B*a*b^2*x^2*sgn(b*x + a) + 1/2*A*b^3*x^2*sgn(b*x + a) + 3*B*a^2*b*x*sgn(b*x +
a) + 3*A*a*b^2*x*sgn(b*x + a) - A*a^3*sgn(b*x + a)/x + (B*a^3*sgn(b*x + a) + 3*A*a^2*b*sgn(b*x + a))*log(abs(
x))

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maple [A]  time = 0.06, size = 96, normalized size = 0.48 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (2 B \,b^{3} x^{4}+3 A \,b^{3} x^{3}+9 B a \,b^{2} x^{3}+18 A \,a^{2} b x \ln \relax (x )+18 A a \,b^{2} x^{2}+6 B \,a^{3} x \ln \relax (x )+18 B \,a^{2} b \,x^{2}-6 A \,a^{3}\right )}{6 \left (b x +a \right )^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^2,x)

[Out]

1/6*((b*x+a)^2)^(3/2)*(2*B*b^3*x^4+3*A*b^3*x^3+9*B*a*b^2*x^3+18*A*ln(x)*x*a^2*b+18*A*a*b^2*x^2+6*B*ln(x)*x*a^3
+18*B*a^2*b*x^2-6*A*a^3)/(b*x+a)^3/x

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maxima [B]  time = 0.54, size = 283, normalized size = 1.42 \begin {gather*} \left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a^{3} \log \left (2 \, b^{2} x + 2 \, a b\right ) + 3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a^{2} b \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) - 3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A a^{2} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a b x + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2} x + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a b + \frac {1}{3} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^2,x, algorithm="maxima")

[Out]

(-1)^(2*b^2*x + 2*a*b)*B*a^3*log(2*b^2*x + 2*a*b) + 3*(-1)^(2*b^2*x + 2*a*b)*A*a^2*b*log(2*b^2*x + 2*a*b) - (-
1)^(2*a*b*x + 2*a^2)*B*a^3*log(2*a*b*x/abs(x) + 2*a^2/abs(x)) - 3*(-1)^(2*a*b*x + 2*a^2)*A*a^2*b*log(2*a*b*x/a
bs(x) + 2*a^2/abs(x)) + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a*b*x + 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b^2*x
+ 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^2 + 9/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*a*b + 1/3*(b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*B - (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/x^2,x)

[Out]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/x**2,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**2, x)

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