Optimal. Leaf size=77 \[ \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{6 a^2 x^6}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7} \]
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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} \begin {gather*} \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{6 a^2 x^6}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7} \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) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}-\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 )^{5/2}}{x^7} \, dx}{2 a b}\\ &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}-\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 )^5}{x^7} \, dx}{2 a b^5 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 a^2 x^6}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 122, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 (6 A+7 B x)+7 a^4 b x (5 A+6 B x)+21 a^3 b^2 x^2 (4 A+5 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+35 a b^4 x^4 (2 A+3 B x)+21 b^5 x^5 (A+2 B x)\right )}{42 x^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.80, size = 780, normalized size = 10.13 \begin {gather*} \frac {32 b^6 \sqrt {a^2+2 a b x+b^2 x^2} \left (-6 a^{11} A b-7 a^{11} b B x-71 a^{10} A b^2 x-84 a^{10} b^2 B x^2-384 a^9 A b^3 x^2-462 a^9 b^3 B x^3-1254 a^8 A b^4 x^3-1540 a^8 b^4 B x^4-2750 a^7 A b^5 x^4-3465 a^7 b^5 B x^5-4257 a^6 A b^6 x^5-5544 a^6 b^6 B x^6-4752 a^5 A b^7 x^6-6461 a^5 b^7 B x^7-3829 a^4 A b^8 x^7-5502 a^4 b^8 B x^8-2184 a^3 A b^9 x^8-3360 a^3 b^9 B x^9-840 a^2 A b^{10} x^9-1400 a^2 b^{10} B x^{10}-196 a A b^{11} x^{10}-357 a b^{11} B x^{11}-21 A b^{12} x^{11}-42 b^{12} B x^{12}\right )+32 \sqrt {b^2} b^6 \left (6 a^{12} A+7 a^{12} B x+77 a^{11} A b x+91 a^{11} b B x^2+455 a^{10} A b^2 x^2+546 a^{10} b^2 B x^3+1638 a^9 A b^3 x^3+2002 a^9 b^3 B x^4+4004 a^8 A b^4 x^4+5005 a^8 b^4 B x^5+7007 a^7 A b^5 x^5+9009 a^7 b^5 B x^6+9009 a^6 A b^6 x^6+12005 a^6 b^6 B x^7+8581 a^5 A b^7 x^7+11963 a^5 b^7 B x^8+6013 a^4 A b^8 x^8+8862 a^4 b^8 B x^9+3024 a^3 A b^9 x^9+4760 a^3 b^9 B x^{10}+1036 a^2 A b^{10} x^{10}+1757 a^2 b^{10} B x^{11}+217 a A b^{11} x^{11}+399 a b^{11} B x^{12}+21 A b^{12} x^{12}+42 b^{12} B x^{13}\right )}{21 \sqrt {b^2} x^7 \sqrt {a^2+2 a b x+b^2 x^2} \left (-64 a^6 b^6-384 a^5 b^7 x-960 a^4 b^8 x^2-1280 a^3 b^9 x^3-960 a^2 b^{10} x^4-384 a b^{11} x^5-64 b^{12} x^6\right )+21 x^7 \left (64 a^7 b^7+448 a^6 b^8 x+1344 a^5 b^9 x^2+2240 a^4 b^{10} x^3+2240 a^3 b^{11} x^4+1344 a^2 b^{12} x^5+448 a b^{13} x^6+64 b^{14} x^7\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 119, normalized size = 1.55 \begin {gather*} -\frac {42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 221, normalized size = 2.87 \begin {gather*} -\frac {{\left (7 \, B a b^{6} - A b^{7}\right )} \mathrm {sgn}\left (b x + a\right )}{42 \, a^{2}} - \frac {42 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 105 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 21 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 140 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 70 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 105 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 42 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 35 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 6 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{42 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 140, normalized size = 1.82 \begin {gather*} -\frac {\left (42 B \,b^{5} x^{6}+21 A \,b^{5} x^{5}+105 B a \,b^{4} x^{5}+70 A a \,b^{4} x^{4}+140 B \,a^{2} b^{3} x^{4}+105 A \,a^{2} b^{3} x^{3}+105 B \,a^{3} b^{2} x^{3}+84 A \,a^{3} b^{2} x^{2}+42 B \,a^{4} b \,x^{2}+35 A \,a^{4} b x +7 B \,a^{5} x +6 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 435, normalized size = 5.65 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{6 \, a^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{6 \, a^{7}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{6 \, a^{5} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{6 \, a^{6} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{6 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{6 \, a^{4} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{6 \, a^{2} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{7 \, a^{2} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 284, normalized size = 3.69 \begin {gather*} -\frac {\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^2\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x\,\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}}{3\,x^3\,\left (a+b\,x\right )}-\frac {a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^4\,\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^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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