Optimal. Leaf size=75 \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 75, 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} \begin {gather*} \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rule 769
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}-\frac {\left (2 A b^2-2 a b B\right ) \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx}{2 a b}\\ &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}-\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 {a b+b^2 x}{x^3} \, dx}{2 a b \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (a (2 A+3 B x)+3 b x (A+2 B x))}{6 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 6.41, size = 403, normalized size = 5.37 \begin {gather*} \frac {2 b^2 (a+b x)^2 (a+2 b x)^{13} \left (2 a A+3 a B x+3 A b x+6 b B x^2\right )}{3 x^3 \sqrt {a^2+2 a b x+b^2 x^2} \left (-2 a^{14} b^2-54 a^{13} b^3 x-676 a^{12} b^4 x^2-5200 a^{11} b^5 x^3-27456 a^{10} b^6 x^4-105248 a^9 b^7 x^5-302016 a^8 b^8 x^6-658944 a^7 b^9 x^7-1098240 a^6 b^{10} x^8-1391104 a^5 b^{11} x^9-1317888 a^4 b^{12} x^{10}-905216 a^3 b^{13} x^{11}-425984 a^2 b^{14} x^{12}-122880 a b^{15} x^{13}-16384 b^{16} x^{14}\right )+3 \sqrt {b^2} x^3 \left (2 a^{15} b+56 a^{14} b^2 x+730 a^{13} b^3 x^2+5876 a^{12} b^4 x^3+32656 a^{11} b^5 x^4+132704 a^{10} b^6 x^5+407264 a^9 b^7 x^6+960960 a^8 b^8 x^7+1757184 a^7 b^9 x^8+2489344 a^6 b^{10} x^9+2708992 a^5 b^{11} x^{10}+2223104 a^4 b^{12} x^{11}+1331200 a^3 b^{13} x^{12}+548864 a^2 b^{14} x^{13}+139264 a b^{15} x^{14}+16384 b^{16} x^{15}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 27, normalized size = 0.36 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 1.03 \begin {gather*} -\frac {{\left (3 \, B a b^{2} - A b^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac {6 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a x \mathrm {sgn}\left (b x + a\right ) + 3 \, A b x \mathrm {sgn}\left (b x + a\right ) + 2 \, A a \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.59 \begin {gather*} -\frac {\left (6 B b \,x^{2}+3 A b x +3 B a x +2 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{6 \left (b x +a \right ) x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 195, normalized size = 2.60 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{2 \, a^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{2 \, a^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{2 \, a x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{2 \, a^{2} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{2 \, a^{2} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 43, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,A\,a+3\,A\,b\,x+3\,B\,a\,x+6\,B\,b\,x^2\right )}{6\,x^3\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 31, normalized size = 0.41 \begin {gather*} \frac {- 2 A a - 6 B b x^{2} + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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