Optimal. Leaf size=160 \[ \frac {32 c^3 \sqrt {b x+c x^2} (9 b B-8 A c)}{315 b^5 x}-\frac {16 c^2 \sqrt {b x+c x^2} (9 b B-8 A c)}{315 b^4 x^2}+\frac {4 c \sqrt {b x+c x^2} (9 b B-8 A c)}{105 b^3 x^3}-\frac {2 \sqrt {b x+c x^2} (9 b B-8 A c)}{63 b^2 x^4}-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5} \]
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Rubi [A] time = 0.14, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 658, 650} \begin {gather*} \frac {32 c^3 \sqrt {b x+c x^2} (9 b B-8 A c)}{315 b^5 x}-\frac {16 c^2 \sqrt {b x+c x^2} (9 b B-8 A c)}{315 b^4 x^2}+\frac {4 c \sqrt {b x+c x^2} (9 b B-8 A c)}{105 b^3 x^3}-\frac {2 \sqrt {b x+c x^2} (9 b B-8 A c)}{63 b^2 x^4}-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx &=-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}+\frac {\left (2 \left (-5 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^4 \sqrt {b x+c x^2}} \, dx}{9 b}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {(2 c (9 b B-8 A c)) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{21 b^2}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}+\frac {\left (8 c^2 (9 b B-8 A c)\right ) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}-\frac {16 c^2 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^4 x^2}-\frac {\left (16 c^3 (9 b B-8 A c)\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{315 b^4}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}-\frac {16 c^2 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^4 x^2}+\frac {32 c^3 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^5 x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 0.62 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (A \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )+9 b B x \left (5 b^3-6 b^2 c x+8 b c^2 x^2-16 c^3 x^3\right )\right )}{315 b^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 108, normalized size = 0.68 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (35 A b^4-40 A b^3 c x+48 A b^2 c^2 x^2-64 A b c^3 x^3+128 A c^4 x^4+45 b^4 B x-54 b^3 B c x^2+72 b^2 B c^2 x^3-144 b B c^3 x^4\right )}{315 b^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 106, normalized size = 0.66 \begin {gather*} -\frac {2 \, {\left (35 \, A b^{4} - 16 \, {\left (9 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 8 \, {\left (9 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 6 \, {\left (9 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} + 5 \, {\left (9 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, b^{5} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 251, normalized size = 1.57 \begin {gather*} \frac {2 \, {\left (630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B c^{\frac {3}{2}} + 756 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b c + 1008 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A c^{2} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} \sqrt {c} + 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{\frac {3}{2}} + 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{3} + 1080 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} \sqrt {c} + 35 \, A b^{4}\right )}}{315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 110, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-144 B b \,c^{3} x^{4}-64 A b \,c^{3} x^{3}+72 B \,b^{2} c^{2} x^{3}+48 A \,b^{2} c^{2} x^{2}-54 B \,b^{3} c \,x^{2}-40 A \,b^{3} c x +45 b^{4} B x +35 A \,b^{4}\right )}{315 \sqrt {c \,x^{2}+b x}\, b^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 198, normalized size = 1.24 \begin {gather*} \frac {32 \, \sqrt {c x^{2} + b x} B c^{3}}{35 \, b^{4} x} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{4}}{315 \, b^{5} x} - \frac {16 \, \sqrt {c x^{2} + b x} B c^{2}}{35 \, b^{3} x^{2}} + \frac {128 \, \sqrt {c x^{2} + b x} A c^{3}}{315 \, b^{4} x^{2}} + \frac {12 \, \sqrt {c x^{2} + b x} B c}{35 \, b^{2} x^{3}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{2}}{105 \, b^{3} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{7 \, b x^{4}} + \frac {16 \, \sqrt {c x^{2} + b x} A c}{63 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{9 \, b x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 146, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c\,x^2+b\,x}\,\left (128\,A\,c^3-144\,B\,b\,c^2\right )}{315\,b^4\,x^2}-\frac {\sqrt {c\,x^2+b\,x}\,\left (256\,A\,c^4-288\,B\,b\,c^3\right )}{315\,b^5\,x}-\frac {\left (32\,A\,c^2-36\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x^3}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{9\,b\,x^5}+\frac {\sqrt {c\,x^2+b\,x}\,\left (16\,A\,c-18\,B\,b\right )}{63\,b^2\,x^4} \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^{5} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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