Optimal. Leaf size=133 \[ -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{1155 a^4 b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6} \]
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Rubi [A] time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{1155 a^4 b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rubi steps
\begin {align*} \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}+\frac {3 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx}{11 a}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}+\frac {2 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx}{33 a^2}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6}+\frac {2 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{231 a^3}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{1155 a^4 b (a+b x)^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 71, normalized size = 0.53 \[ -\frac {(a-b x)^2 \sqrt {a^2-b^2 x^2} \left (152 a^3+61 a^2 b x+16 a b^2 x^2+2 b^3 x^3\right )}{1155 a^4 b (a+b x)^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 203, normalized size = 1.53 \[ -\frac {152 \, b^{6} x^{6} + 912 \, a b^{5} x^{5} + 2280 \, a^{2} b^{4} x^{4} + 3040 \, a^{3} b^{3} x^{3} + 2280 \, a^{4} b^{2} x^{2} + 912 \, a^{5} b x + 152 \, a^{6} + {\left (2 \, b^{5} x^{5} + 12 \, a b^{4} x^{4} + 31 \, a^{2} b^{3} x^{3} + 46 \, a^{3} b^{2} x^{2} - 243 \, a^{4} b x + 152 \, a^{5}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{4} b^{7} x^{6} + 6 \, a^{5} b^{6} x^{5} + 15 \, a^{6} b^{5} x^{4} + 20 \, a^{7} b^{4} x^{3} + 15 \, a^{8} b^{3} x^{2} + 6 \, a^{9} b^{2} x + a^{10} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 351, normalized size = 2.64 \[ \frac {2 \, {\left (\frac {517 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {4895 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {11220 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {27060 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {32802 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {37422 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {23100 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {13860 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac {3465 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac {1155 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + 152\right )}}{1155 \, a^{4} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{11} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 66, normalized size = 0.50 \[ -\frac {\left (-b x +a \right ) \left (2 b^{3} x^{3}+16 a \,b^{2} x^{2}+61 a^{2} b x +152 a^{3}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{1155 \left (b x +a \right )^{7} a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 440, normalized size = 3.31 \[ -\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{4 \, {\left (b^{8} x^{7} + 7 \, a b^{7} x^{6} + 21 \, a^{2} b^{6} x^{5} + 35 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{3} + 21 \, a^{5} b^{3} x^{2} + 7 \, a^{6} b^{2} x + a^{7} b\right )}} + \frac {3 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{22 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{132 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{231 \, {\left (a b^{5} x^{4} + 4 \, a^{2} b^{4} x^{3} + 6 \, a^{3} b^{3} x^{2} + 4 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{385 \, {\left (a^{2} b^{4} x^{3} + 3 \, a^{3} b^{3} x^{2} + 3 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{4} b^{2} x + a^{5} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 170, normalized size = 1.28 \[ \frac {8\,\sqrt {a^2-b^2\,x^2}}{33\,b\,{\left (a+b\,x\right )}^5}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{11\,b\,{\left (a+b\,x\right )}^6}-\frac {\sqrt {a^2-b^2\,x^2}}{231\,a\,b\,{\left (a+b\,x\right )}^4}-\frac {\sqrt {a^2-b^2\,x^2}}{385\,a^2\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{1155\,a^3\,b\,{\left (a+b\,x\right )}^2}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{1155\,a^4\,b\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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