Optimal. Leaf size=163 \[ \frac {128 c^3 (b+2 c x) (9 b B-10 A c)}{315 b^6 \sqrt {b x+c x^2}}-\frac {32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt {b x+c x^2}}+\frac {16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt {b x+c x^2}}-\frac {2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt {b x+c x^2}}-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 163, 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, 613} \[ \frac {128 c^3 (b+2 c x) (9 b B-10 A c)}{315 b^6 \sqrt {b x+c x^2}}-\frac {32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt {b x+c x^2}}+\frac {16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt {b x+c x^2}}-\frac {2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt {b x+c x^2}}-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}}+\frac {\left (2 \left (\frac {1}{2} (b B-2 A c)-4 (-b B+A c)\right )\right ) \int \frac {1}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx}{9 b}\\ &=-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}}-\frac {2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt {b x+c x^2}}-\frac {(8 c (9 b B-10 A c)) \int \frac {1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx}{63 b^2}\\ &=-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}}-\frac {2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt {b x+c x^2}}+\frac {16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt {b x+c x^2}}+\frac {\left (16 c^2 (9 b B-10 A c)\right ) \int \frac {1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{105 b^3}\\ &=-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}}-\frac {2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt {b x+c x^2}}+\frac {16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt {b x+c x^2}}-\frac {32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt {b x+c x^2}}-\frac {\left (64 c^3 (9 b B-10 A c)\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{315 b^4}\\ &=-\frac {2 A}{9 b x^4 \sqrt {b x+c x^2}}-\frac {2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt {b x+c x^2}}+\frac {16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt {b x+c x^2}}-\frac {32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt {b x+c x^2}}+\frac {128 c^3 (9 b B-10 A c) (b+2 c x)}{315 b^6 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 123, normalized size = 0.75 \[ -\frac {2 \left (5 A \left (7 b^5-10 b^4 c x+16 b^3 c^2 x^2-32 b^2 c^3 x^3+128 b c^4 x^4+256 c^5 x^5\right )+9 b B x \left (5 b^4-8 b^3 c x+16 b^2 c^2 x^2-64 b c^3 x^3-128 c^4 x^4\right )\right )}{315 b^6 x^4 \sqrt {x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 142, normalized size = 0.87 \[ -\frac {2 \, {\left (35 \, A b^{5} - 128 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \, {\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 16 \, {\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 8 \, {\left (9 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 5 \, {\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, {\left (b^{6} c x^{6} + b^{7} x^{5}\right )}} \]
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 \[ \int \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 134, normalized size = 0.82 \[ -\frac {2 \left (c x +b \right ) \left (1280 A \,c^{5} x^{5}-1152 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-576 B \,b^{2} c^{3} x^{4}-160 A \,b^{2} c^{3} x^{3}+144 B \,b^{3} c^{2} x^{3}+80 A \,b^{3} c^{2} x^{2}-72 B \,b^{4} c \,x^{2}-50 A \,b^{4} c x +45 B \,b^{5} x +35 A \,b^{5}\right )}{315 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 234, normalized size = 1.44 \[ \frac {256 \, B c^{4} x}{35 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {512 \, A c^{5} x}{63 \, \sqrt {c x^{2} + b x} b^{6}} + \frac {128 \, B c^{3}}{35 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {256 \, A c^{4}}{63 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {32 \, B c^{2}}{35 \, \sqrt {c x^{2} + b x} b^{3} x} + \frac {64 \, A c^{3}}{63 \, \sqrt {c x^{2} + b x} b^{4} x} + \frac {16 \, B c}{35 \, \sqrt {c x^{2} + b x} b^{2} x^{2}} - \frac {32 \, A c^{2}}{63 \, \sqrt {c x^{2} + b x} b^{3} x^{2}} - \frac {2 \, B}{7 \, \sqrt {c x^{2} + b x} b x^{3}} + \frac {20 \, A c}{63 \, \sqrt {c x^{2} + b x} b^{2} x^{3}} - \frac {2 \, A}{9 \, \sqrt {c x^{2} + b x} b x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 191, normalized size = 1.17 \[ \frac {\sqrt {c\,x^2+b\,x}\,\left (x\,\left (\frac {1300\,A\,c^5-1044\,B\,b\,c^4}{315\,b^6}-\frac {4\,c^4\,\left (965\,A\,c-837\,B\,b\right )}{315\,b^6}\right )-\frac {2\,c^3\,\left (965\,A\,c-837\,B\,b\right )}{315\,b^5}\right )}{x\,\left (b+c\,x\right )}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{9\,b^2\,x^5}-\frac {\left (18\,B\,b^2-34\,A\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{63\,b^4\,x^4}-\frac {2\,c\,\sqrt {c\,x^2+b\,x}\,\left (55\,A\,c-39\,B\,b\right )}{105\,b^4\,x^3}+\frac {2\,c^2\,\sqrt {c\,x^2+b\,x}\,\left (325\,A\,c-261\,B\,b\right )}{315\,b^5\,x^2} \]
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 \[ \int \frac {A + B x}{x^{4} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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