Optimal. Leaf size=69 \[ \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A b-a B)}{4 b^2}+\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {640, 609} \begin {gather*} \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A b-a B)}{4 b^2}+\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 640
Rubi steps
\begin {align*} \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}+\frac {\left (2 A b^2-2 a b B\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx}{2 b^2}\\ &=\frac {(A b-a B) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2}+\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 83, normalized size = 1.20 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (10 a^3 (2 A+B x)+10 a^2 b x (3 A+2 B x)+5 a b^2 x^2 (4 A+3 B x)+b^3 x^3 (5 A+4 B x)\right )}{20 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.87, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 69, normalized size = 1.00 \begin {gather*} \frac {1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + {\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 144, normalized size = 2.09 \begin {gather*} \frac {1}{5} \, B b^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a b^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + B a^{2} b x^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{3} x \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (B a^{5} - 5 \, A a^{4} b\right )} \mathrm {sgn}\left (b x + a\right )}{20 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 90, normalized size = 1.30 \begin {gather*} \frac {\left (4 b^{3} B \,x^{4}+5 A \,b^{3} x^{3}+15 x^{3} B a \,b^{2}+20 x^{2} A a \,b^{2}+20 B \,a^{2} b \,x^{2}+30 x A \,a^{2} b +10 B \,a^{3} x +20 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{20 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 125, normalized size = 1.81 \begin {gather*} \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a x}{4 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{5 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 42, normalized size = 0.61 \begin {gather*} \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}\,\left (5\,A\,b-B\,a+4\,B\,b\,x\right )}{20\,b^2} \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 \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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