Optimal. Leaf size=77 \[ \frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{4 a^2 x^4}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {769, 646, 37} \[ \frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{4 a^2 x^4}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rule 769
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5}-\frac {\left (2 A b^2-2 a b B\right ) \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^5} \, dx}{2 a b}\\ &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5}-\frac {\left (\left (2 A b^2-2 a b B\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^3}{x^5} \, dx}{2 a b^3 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 a^2 x^4}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 84, normalized size = 1.09 \[ -\frac {\sqrt {(a+b x)^2} \left (a^3 (4 A+5 B x)+5 a^2 b x (3 A+4 B x)+10 a b^2 x^2 (2 A+3 B x)+10 b^3 x^3 (A+2 B x)\right )}{20 x^5 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 73, normalized size = 0.95 \[ -\frac {20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 149, normalized size = 1.94 \[ -\frac {{\left (5 \, B a b^{4} - A b^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{20 \, a^{2}} - \frac {20 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 30 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{20 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 1.19 \[ -\frac {\left (20 B \,b^{3} x^{4}+10 A \,b^{3} x^{3}+30 B a \,b^{2} x^{3}+20 A a \,b^{2} x^{2}+20 B \,a^{2} b \,x^{2}+15 A \,a^{2} b x +5 B \,a^{3} x +4 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 \left (b x +a \right )^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.81, size = 315, normalized size = 4.09 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{4}}{4 \, a^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{5}}{4 \, a^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{3}}{4 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{4}}{4 \, a^{4} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{4 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{4 \, a^{5} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{4 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{4 \, a^{4} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{4 \, a^{2} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{4 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{5 \, a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 196, normalized size = 2.55 \[ -\frac {\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^4\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{2}+\frac {3\,B\,a\,b^2}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^2\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x\,\left (a+b\,x\right )}-\frac {a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )} \]
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 \[ \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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