Optimal. Leaf size=200 \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{x (a+b x)}+\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}+\frac {3 a b \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {b^3 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
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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 {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{x (a+b x)}+\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}+\frac {3 a b \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac {b^3 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (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^3} \, 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^3} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^5 (A b+3 a B)+\frac {a^3 A b^3}{x^3}+\frac {a^2 b^3 (3 A b+a B)}{x^2}+\frac {3 a b^4 (A b+a B)}{x}+b^6 B x\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}}{2 x^2 (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^2 (A b+3 a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a b (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 = 0.04, size = 85, normalized size = 0.42 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-\left (a^3 (A+2 B x)\right )-6 a^2 A b x+6 a b x^2 \log (x) (a B+A b)+6 a b^2 B x^3+b^3 x^3 (2 A+B x)\right )}{2 x^2 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.58, size = 1061, normalized size = 5.30 \begin {gather*} 3 a A \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right ) b^2-\frac {3}{2} a A \sqrt {b^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) b-\frac {3}{2} a A \sqrt {b^2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) b+\frac {-A \sqrt {a^2+2 b x a+b^2 x^2} \left (-2 x^3 b^4-a x^2 b^3+6 a^2 x b^2+a^3 b\right )-A \sqrt {b^2} \left (-a^4-7 b x a^3-5 b^2 x^2 a^2+3 b^3 x^3 a+2 b^4 x^4\right )}{2 x^2 \left (x b^2+a b\right )-2 \sqrt {b^2} x^2 \sqrt {a^2+2 b x a+b^2 x^2}}+\frac {2 \sqrt {b^2} B a^4-\frac {3}{4} b \sqrt {b^2} B x a^3-6 b^2 B x \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right ) a^3-3 b \sqrt {b^2} B x \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) a^3-3 b \sqrt {b^2} B x \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) a^3-2 b B \sqrt {a^2+2 b x a+b^2 x^2} a^3-\frac {35}{4} \left (b^2\right )^{3/2} B x^2 a^2-6 b^3 B x^2 \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right ) a^2+6 b \sqrt {b^2} B x \sqrt {a^2+2 b x a+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right ) a^2-3 \left (b^2\right )^{3/2} B x^2 \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) a^2+3 b^2 B x \sqrt {a^2+2 b x a+b^2 x^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) a^2-3 \left (b^2\right )^{3/2} B x^2 \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) a^2+3 b^2 B x \sqrt {a^2+2 b x a+b^2 x^2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) a^2+\frac {11}{4} b^2 B x \sqrt {a^2+2 b x a+b^2 x^2} a^2-7 b^3 \sqrt {b^2} B x^3 a+6 b^3 B x^2 \sqrt {a^2+2 b x a+b^2 x^2} a-b^4 \sqrt {b^2} B x^4+b^4 B x^3 \sqrt {a^2+2 b x a+b^2 x^2}}{\left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 74, normalized size = 0.37 \begin {gather*} \frac {B b^{3} x^{4} - A a^{3} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} \log \relax (x) - 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 117, normalized size = 0.58 \begin {gather*} \frac {1}{2} \, B b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} x \mathrm {sgn}\left (b x + a\right ) + A b^{3} x \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (B a^{2} b \mathrm {sgn}\left (b x + a\right ) + A a b^{2} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (B a^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{2} b \mathrm {sgn}\left (b x + a\right )\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 95, normalized size = 0.48 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (B \,b^{3} x^{4}+6 A a \,b^{2} x^{2} \ln \relax (x )+2 A \,b^{3} x^{3}+6 B \,a^{2} b \,x^{2} \ln \relax (x )+6 B a \,b^{2} x^{3}-6 A \,a^{2} b x -2 B \,a^{3} x -A \,a^{3}\right )}{2 \left (b x +a \right )^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 351, normalized size = 1.76 \begin {gather*} 3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a^{2} b \log \left (2 \, b^{2} x + 2 \, a b\right ) + 3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a b^{2} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a^{2} b \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 b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2} x + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3} x}{2 \, a} + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a b + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2}}{2 \, a^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{2 \, a x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{2 \, a^{2} x^{2}} \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^3} \,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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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