Optimal. Leaf size=130 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{28 a^2 x^7}-\frac {A \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{168 a^3 x^6} \]
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Rubi [A] time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {770, 78, 45, 37} \begin {gather*} -\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{168 a^3 x^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{28 a^2 x^7}-\frac {A \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {A (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {\left (\left (-2 A b^2+8 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 )^5}{x^8} \, dx}{8 a b^5 \left (a b+b^2 x\right )}\\ &=-\frac {A (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}-\frac {\left (\left (-2 A b^2+8 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 )^5}{x^7} \, dx}{56 a^2 b^4 \left (a b+b^2 x\right )}\\ &=-\frac {A (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}-\frac {b (A b-4 a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{168 a^3 x^6}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 125, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a^5 (7 A+8 B x)+20 a^4 b x (6 A+7 B x)+56 a^3 b^2 x^2 (5 A+6 B x)+84 a^2 b^3 x^3 (4 A+5 B x)+70 a b^4 x^4 (3 A+4 B x)+28 b^5 x^5 (2 A+3 B x)\right )}{168 x^8 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 3.29, size = 850, normalized size = 6.54 \begin {gather*} \frac {16 \sqrt {a^2+2 b x a+b^2 x^2} \left (-84 B x^{13} b^{13}-56 A x^{12} b^{13}-868 a B x^{12} b^{12}-602 a A x^{11} b^{12}-4144 a^2 B x^{11} b^{11}-2982 a^2 A x^{10} b^{11}-12096 a^3 B x^{10} b^{10}-9002 a^3 A x^9 b^{10}-24052 a^4 B x^9 b^9-18446 a^4 A x^8 b^9-34324 a^5 B x^8 b^8-27027 a^5 A x^7 b^8-36036 a^6 B x^7 b^7-29029 a^6 A x^6 b^7-28028 a^7 B x^6 b^6-23023 a^7 A x^5 b^6-16016 a^8 B x^5 b^5-13377 a^8 A x^4 b^5-6552 a^9 B x^4 b^4-5551 a^9 A x^3 b^4-1820 a^{10} B x^3 b^3-1561 a^{10} A x^2 b^3-308 a^{11} B x^2 b^2-267 a^{11} A x b^2-21 a^{12} A b-24 a^{12} B x b\right ) b^7+16 \sqrt {b^2} \left (84 b^{13} B x^{14}+56 A b^{13} x^{13}+952 a b^{12} B x^{13}+658 a A b^{12} x^{12}+5012 a^2 b^{11} B x^{12}+3584 a^2 A b^{11} x^{11}+16240 a^3 b^{10} B x^{11}+11984 a^3 A b^{10} x^{10}+36148 a^4 b^9 B x^{10}+27448 a^4 A b^9 x^9+58376 a^5 b^8 B x^9+45473 a^5 A b^8 x^8+70360 a^6 b^7 B x^8+56056 a^6 A b^7 x^7+64064 a^7 b^6 B x^7+52052 a^7 A b^6 x^6+44044 a^8 b^5 B x^6+36400 a^8 A b^5 x^5+22568 a^9 b^4 B x^5+18928 a^9 A b^4 x^4+8372 a^{10} b^3 B x^4+7112 a^{10} A b^3 x^3+2128 a^{11} b^2 B x^3+1828 a^{11} A b^2 x^2+332 a^{12} b B x^2+288 a^{12} A b x+24 a^{13} B x+21 a^{13} A\right ) b^7}{21 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-128 x^7 b^{14}-896 a x^6 b^{13}-2688 a^2 x^5 b^{12}-4480 a^3 x^4 b^{11}-4480 a^4 x^3 b^{10}-2688 a^5 x^2 b^9-896 a^6 x b^8-128 a^7 b^7\right ) x^8+21 \left (128 x^8 b^{16}+1024 a x^7 b^{15}+3584 a^2 x^6 b^{14}+7168 a^3 x^5 b^{13}+8960 a^4 x^4 b^{12}+7168 a^5 x^3 b^{11}+3584 a^6 x^2 b^{10}+1024 a^7 x b^9+128 a^8 b^8\right ) x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 119, normalized size = 0.92 \begin {gather*} -\frac {84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 221, normalized size = 1.70 \begin {gather*} \frac {{\left (4 \, B a b^{7} - A b^{8}\right )} \mathrm {sgn}\left (b x + a\right )}{168 \, a^{3}} - \frac {84 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 280 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 56 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 420 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 210 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 336 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 336 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 140 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 280 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 24 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 120 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 21 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 140, normalized size = 1.08 \begin {gather*} -\frac {\left (84 B \,b^{5} x^{6}+56 A \,b^{5} x^{5}+280 B a \,b^{4} x^{5}+210 A a \,b^{4} x^{4}+420 B \,a^{2} b^{3} x^{4}+336 A \,a^{2} b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+280 A \,a^{3} b^{2} x^{2}+140 B \,a^{4} b \,x^{2}+120 A \,a^{4} b x +24 B \,a^{5} x +21 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 495, normalized size = 3.81 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{6 \, a^{7}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{6 \, a^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{6 \, a^{6} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{6 \, a^{7} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{6 \, a^{4} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{7 \, a^{2} x^{7}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{56 \, a^{3} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{8 \, a^{2} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 284, normalized size = 2.18 \begin {gather*} -\frac {\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^7\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {2\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )} \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 {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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