Optimal. Leaf size=210 \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 x^6 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{4 x^4 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]
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Rubi [A] time = 0.08, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 x^6 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{4 x^4 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^8} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{x^8} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 A b^3}{x^8}+\frac {a^2 b^3 (3 A b+a B)}{x^7}+\frac {3 a b^4 (A b+a B)}{x^6}+\frac {b^5 (A b+3 a B)}{x^5}+\frac {b^6 B}{x^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {3 a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 0.41 \[ -\frac {\sqrt {(a+b x)^2} \left (10 a^3 (6 A+7 B x)+42 a^2 b x (5 A+6 B x)+63 a b^2 x^2 (4 A+5 B x)+35 b^3 x^3 (3 A+4 B x)\right )}{420 x^7 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 73, normalized size = 0.35 \[ -\frac {140 \, B b^{3} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 149, normalized size = 0.71 \[ -\frac {{\left (7 \, B a b^{6} - 3 \, A b^{7}\right )} \mathrm {sgn}\left (b x + a\right )}{420 \, a^{4}} - \frac {140 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 315 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 252 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 252 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 210 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 60 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{420 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 0.44 \[ -\frac {\left (140 B \,b^{3} x^{4}+105 A \,b^{3} x^{3}+315 B a \,b^{2} x^{3}+252 A a \,b^{2} x^{2}+252 B \,a^{2} b \,x^{2}+210 A \,a^{2} b x +70 B \,a^{3} x +60 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{420 \left (b x +a \right )^{3} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 435, normalized size = 2.07 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{6}}{4 \, a^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{7}}{4 \, a^{7}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{5}}{4 \, a^{5} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{6}}{4 \, a^{6} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{4}}{4 \, a^{6} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{5}}{4 \, a^{7} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{4 \, a^{5} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{4 \, a^{6} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{4 \, a^{4} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{4 \, a^{5} x^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{30 \, a^{3} x^{5}} - \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{70 \, a^{4} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{6 \, a^{2} x^{6}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{14 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{7 \, a^{2} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 196, normalized size = 0.93 \[ -\frac {\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{4}+\frac {3\,B\,a\,b^2}{4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^4\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )} \]
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 \[ \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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