Optimal. Leaf size=70 \[ -\frac {8 (b+2 c x) (b B-2 A c)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {638, 613} \begin {gather*} -\frac {8 (b+2 c x) (b B-2 A c)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 638
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (A b-(b B-2 A c) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {(4 (b B-2 A c)) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (A b-(b B-2 A c) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {8 (b B-2 A c) (b+2 c x)}{3 b^4 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 1.03 \begin {gather*} -\frac {2 \left (A \left (b^3-6 b^2 c x-24 b c^2 x^2-16 c^3 x^3\right )+b B x \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 90, normalized size = 1.29 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (A b^3-6 A b^2 c x-24 A b c^2 x^2-16 A c^3 x^3+3 b^3 B x+12 b^2 B c x^2+8 b B c^2 x^3\right )}{3 b^4 x^2 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 101, normalized size = 1.44 \begin {gather*} -\frac {2 \, {\left (A b^{3} + 8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 12 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 82, normalized size = 1.17 \begin {gather*} -\frac {2 \, {\left ({\left (4 \, x {\left (\frac {2 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} x}{b^{4}} + \frac {3 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )}}{b^{4}}\right )} + \frac {3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )}}{b^{4}}\right )} x + \frac {A}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 1.19 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-16 A \,c^{3} x^{3}+8 B b \,c^{2} x^{3}-24 A b \,c^{2} x^{2}+12 B \,b^{2} c \,x^{2}-6 A \,b^{2} c x +3 B \,b^{3} x +A \,b^{3}\right ) x}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 130, normalized size = 1.86 \begin {gather*} \frac {2 \, B x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, B c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {4 \, A c x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {8 \, B}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, A}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c}{3 \, \sqrt {c x^{2} + b x} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 76, normalized size = 1.09 \begin {gather*} -\frac {2\,\left (3\,B\,b^3\,x+A\,b^3+12\,B\,b^2\,c\,x^2-6\,A\,b^2\,c\,x+8\,B\,b\,c^2\,x^3-24\,A\,b\,c^2\,x^2-16\,A\,c^3\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \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 \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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