Optimal. Leaf size=170 \[ \frac {16 c^3 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{15015 b^5 x^{10}}-\frac {8 c^2 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{3003 b^4 x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{429 b^3 x^{14}}-\frac {\left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{143 b^2 x^{16}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}} \]
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Rubi [A] time = 0.32, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \[ \frac {16 c^3 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{15015 b^5 x^{10}}-\frac {8 c^2 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{3003 b^4 x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{429 b^3 x^{14}}-\frac {\left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{143 b^2 x^{16}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}} \]
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rule 792
Rule 2034
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{17}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^9} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}+\frac {\left (-9 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^8} \, dx,x,x^2\right )}{13 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac {(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}-\frac {(3 c (13 b B-8 A c)) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )}{143 b^2}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac {(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}+\frac {2 c (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}+\frac {\left (4 c^2 (13 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{429 b^3}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac {(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}+\frac {2 c (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}-\frac {8 c^2 (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{3003 b^4 x^{12}}-\frac {\left (8 c^3 (13 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{3003 b^4}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac {(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}+\frac {2 c (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}-\frac {8 c^2 (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{3003 b^4 x^{12}}+\frac {16 c^3 (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{15015 b^5 x^{10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 95, normalized size = 0.56 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\left (\frac {c x^3}{b}+x\right )^2 \left (105 b^3-70 b^2 c x^2+40 b c^2 x^4-16 c^3 x^6\right ) (8 A c-13 b B)-1155 A b^2 \left (b+c x^2\right )^2\right )}{15015 b^3 x^{14}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 157, normalized size = 0.92 \[ \frac {{\left (16 \, {\left (13 \, B b c^{5} - 8 \, A c^{6}\right )} x^{12} - 8 \, {\left (13 \, B b^{2} c^{4} - 8 \, A b c^{5}\right )} x^{10} + 6 \, {\left (13 \, B b^{3} c^{3} - 8 \, A b^{2} c^{4}\right )} x^{8} - 1155 \, A b^{6} - 5 \, {\left (13 \, B b^{4} c^{2} - 8 \, A b^{3} c^{3}\right )} x^{6} - 35 \, {\left (52 \, B b^{5} c + A b^{4} c^{2}\right )} x^{4} - 105 \, {\left (13 \, B b^{6} + 14 \, A b^{5} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15015 \, b^{5} x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.41, size = 550, normalized size = 3.24 \[ \frac {32 \, {\left (15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{18} B c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 3003 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{16} B b c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 48048 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{16} A c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 6006 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} B b^{2} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 96096 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} A b c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 28314 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B b^{3} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 109824 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} A b^{2} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 13728 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b^{4} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 37752 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A b^{3} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 5720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{5} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 5720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b^{4} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 3718 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{6} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 2288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{5} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 1014 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{7} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 624 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{6} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 169 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{8} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 104 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{7} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 13 \, B b^{9} c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 8 \, A b^{8} c^{\frac {13}{2}} \mathrm {sgn}\relax (x)\right )}}{15015 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 118, normalized size = 0.69 \[ -\frac {\left (c \,x^{2}+b \right ) \left (128 A \,c^{4} x^{8}-208 B b \,c^{3} x^{8}-320 A b \,c^{3} x^{6}+520 B \,b^{2} c^{2} x^{6}+560 A \,b^{2} c^{2} x^{4}-910 B \,b^{3} c \,x^{4}-840 A \,b^{3} c \,x^{2}+1365 B \,b^{4} x^{2}+1155 A \,b^{4}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{15015 b^{5} x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.62, size = 337, normalized size = 1.98 \[ \frac {1}{9240} \, B {\left (\frac {128 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{b^{4} x^{2}} - \frac {64 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{3} x^{4}} + \frac {48 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{6}} - \frac {40 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{8}} + \frac {35 \, \sqrt {c x^{4} + b x^{2}} c}{x^{10}} + \frac {315 \, \sqrt {c x^{4} + b x^{2}} b}{x^{12}} - \frac {1155 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{14}}\right )} - \frac {1}{30030} \, A {\left (\frac {256 \, \sqrt {c x^{4} + b x^{2}} c^{6}}{b^{5} x^{2}} - \frac {128 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{b^{4} x^{4}} + \frac {96 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{3} x^{6}} - \frac {80 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{8}} + \frac {70 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{10}} - \frac {63 \, \sqrt {c x^{4} + b x^{2}} c}{x^{12}} - \frac {693 \, \sqrt {c x^{4} + b x^{2}} b}{x^{14}} + \frac {3003 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{16}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 306, normalized size = 1.80 \[ \frac {8\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{3003\,b^2\,x^8}-\frac {14\,A\,c\,\sqrt {c\,x^4+b\,x^2}}{143\,x^{12}}-\frac {B\,b\,\sqrt {c\,x^4+b\,x^2}}{11\,x^{12}}-\frac {4\,B\,c\,\sqrt {c\,x^4+b\,x^2}}{33\,x^{10}}-\frac {A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{429\,b\,x^{10}}-\frac {A\,b\,\sqrt {c\,x^4+b\,x^2}}{13\,x^{14}}-\frac {16\,A\,c^4\,\sqrt {c\,x^4+b\,x^2}}{5005\,b^3\,x^6}+\frac {64\,A\,c^5\,\sqrt {c\,x^4+b\,x^2}}{15015\,b^4\,x^4}-\frac {128\,A\,c^6\,\sqrt {c\,x^4+b\,x^2}}{15015\,b^5\,x^2}-\frac {B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{231\,b\,x^8}+\frac {2\,B\,c^3\,\sqrt {c\,x^4+b\,x^2}}{385\,b^2\,x^6}-\frac {8\,B\,c^4\,\sqrt {c\,x^4+b\,x^2}}{1155\,b^3\,x^4}+\frac {16\,B\,c^5\,\sqrt {c\,x^4+b\,x^2}}{1155\,b^4\,x^2} \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{17}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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