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.
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Rule 76
Rule 770
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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