Optimal. Leaf size=96 \[ \frac {16 c (b+2 c x) (5 b B-8 A c)}{15 b^5 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x) (5 b B-8 A c)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 A}{5 b x \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 614, 613} \begin {gather*} \frac {16 c (b+2 c x) (5 b B-8 A c)}{15 b^5 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x) (5 b B-8 A c)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 A}{5 b x \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 A}{5 b x \left (b x+c x^2\right )^{3/2}}+\frac {\left (2 \left (b B-A c-\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{\left (b x+c x^2\right )^{5/2}} \, dx}{5 b}\\ &=-\frac {2 A}{5 b x \left (b x+c x^2\right )^{3/2}}-\frac {2 (5 b B-8 A c) (b+2 c x)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {(8 c (5 b B-8 A c)) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{15 b^3}\\ &=-\frac {2 A}{5 b x \left (b x+c x^2\right )^{3/2}}-\frac {2 (5 b B-8 A c) (b+2 c x)}{15 b^3 \left (b x+c x^2\right )^{3/2}}+\frac {16 c (5 b B-8 A c) (b+2 c x)}{15 b^5 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 98, normalized size = 1.02 \begin {gather*} -\frac {2 \left (A \left (3 b^4-8 b^3 c x+48 b^2 c^2 x^2+192 b c^3 x^3+128 c^4 x^4\right )+5 b B x \left (b^3-6 b^2 c x-24 b c^2 x^2-16 c^3 x^3\right )\right )}{15 b^5 x (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 115, normalized size = 1.20 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-3 A b^4+8 A b^3 c x-48 A b^2 c^2 x^2-192 A b c^3 x^3-128 A c^4 x^4-5 b^4 B x+30 b^3 B c x^2+120 b^2 B c^2 x^3+80 b B c^3 x^4\right )}{15 b^5 x^3 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 128, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (3 \, A b^{4} - 16 \, {\left (5 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} - 24 \, {\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 6 \, {\left (5 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} + {\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, {\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 1.11 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-80 B b \,c^{3} x^{4}+192 A b \,c^{3} x^{3}-120 B \,b^{2} c^{2} x^{3}+48 A \,b^{2} c^{2} x^{2}-30 B \,b^{3} c \,x^{2}-8 A \,b^{3} c x +5 b^{4} B x +3 A \,b^{4}\right )}{15 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 176, normalized size = 1.83 \begin {gather*} -\frac {4 \, B c x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, B c^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {32 \, A c^{2} x}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}} - \frac {256 \, A c^{3} x}{15 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {2 \, B}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, B c}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {16 \, A c}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} - \frac {128 \, A c^{2}}{15 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {2 \, A}{5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 111, normalized size = 1.16 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (5\,B\,b^4\,x+3\,A\,b^4-30\,B\,b^3\,c\,x^2-8\,A\,b^3\,c\,x-120\,B\,b^2\,c^2\,x^3+48\,A\,b^2\,c^2\,x^2-80\,B\,b\,c^3\,x^4+192\,A\,b\,c^3\,x^3+128\,A\,c^4\,x^4\right )}{15\,b^5\,x^3\,{\left (b+c\,x\right )}^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}{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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