Optimal. Leaf size=447 \[ -\frac {4 b^{17/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-21 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}+\frac {8 b^{17/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-21 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {8 b^4 x^{3/2} \left (b+c x^2\right ) (11 b B-21 A c)}{3315 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {8 b^3 \sqrt {x} \sqrt {b x^2+c x^4} (11 b B-21 A c)}{9945 c^3}-\frac {8 b^2 x^{5/2} \sqrt {b x^2+c x^4} (11 b B-21 A c)}{13923 c^2}-\frac {4 b x^{9/2} \sqrt {b x^2+c x^4} (11 b B-21 A c)}{1547 c}-\frac {2 x^{5/2} \left (b x^2+c x^4\right )^{3/2} (11 b B-21 A c)}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c} \]
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Rubi [A] time = 0.59, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2039, 2021, 2024, 2032, 329, 305, 220, 1196} \[ -\frac {8 b^2 x^{5/2} \sqrt {b x^2+c x^4} (11 b B-21 A c)}{13923 c^2}-\frac {8 b^4 x^{3/2} \left (b+c x^2\right ) (11 b B-21 A c)}{3315 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {8 b^3 \sqrt {x} \sqrt {b x^2+c x^4} (11 b B-21 A c)}{9945 c^3}-\frac {4 b^{17/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-21 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}+\frac {8 b^{17/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-21 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {4 b x^{9/2} \sqrt {b x^2+c x^4} (11 b B-21 A c)}{1547 c}-\frac {2 x^{5/2} \left (b x^2+c x^4\right )^{3/2} (11 b B-21 A c)}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2021
Rule 2024
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {\left (2 \left (\frac {11 b B}{2}-\frac {21 A c}{2}\right )\right ) \int x^{3/2} \left (b x^2+c x^4\right )^{3/2} \, dx}{21 c}\\ &=-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(2 b (11 b B-21 A c)) \int x^{7/2} \sqrt {b x^2+c x^4} \, dx}{119 c}\\ &=-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {\left (4 b^2 (11 b B-21 A c)\right ) \int \frac {x^{11/2}}{\sqrt {b x^2+c x^4}} \, dx}{1547 c}\\ &=-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}+\frac {\left (4 b^3 (11 b B-21 A c)\right ) \int \frac {x^{7/2}}{\sqrt {b x^2+c x^4}} \, dx}{1989 c^2}\\ &=\frac {8 b^3 (11 b B-21 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{9945 c^3}-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {\left (4 b^4 (11 b B-21 A c)\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{3315 c^3}\\ &=\frac {8 b^3 (11 b B-21 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{9945 c^3}-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {\left (4 b^4 (11 b B-21 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{3315 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {8 b^3 (11 b B-21 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{9945 c^3}-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {\left (8 b^4 (11 b B-21 A c) x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{3315 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {8 b^3 (11 b B-21 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{9945 c^3}-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {\left (8 b^{9/2} (11 b B-21 A c) x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{3315 c^{7/2} \sqrt {b x^2+c x^4}}+\frac {\left (8 b^{9/2} (11 b B-21 A c) x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{3315 c^{7/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {8 b^4 (11 b B-21 A c) x^{3/2} \left (b+c x^2\right )}{3315 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {8 b^3 (11 b B-21 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{9945 c^3}-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}+\frac {8 b^{17/4} (11 b B-21 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {4 b^{17/4} (11 b B-21 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 138, normalized size = 0.31 \[ \frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left (7 b^3 (21 A c-11 b B) \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )+\left (b+c x^2\right )^2 \sqrt {\frac {c x^2}{b}+1} \left (-b c \left (147 A+143 B x^2\right )+13 c^2 x^2 \left (21 A+17 B x^2\right )+77 b^2 B\right )\right )}{4641 c^3 \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B c x^{7} + {\left (B b + A c\right )} x^{5} + A b x^{3}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} x^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 494, normalized size = 1.11 \[ \frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3315 B \,c^{6} x^{12}+4095 A \,c^{6} x^{10}+7800 B b \,c^{5} x^{10}+10080 A b \,c^{5} x^{8}+4665 B \,b^{2} c^{4} x^{8}+6405 A \,b^{2} c^{4} x^{6}-40 B \,b^{3} c^{3} x^{6}-168 A \,b^{3} c^{3} x^{4}+88 B \,b^{4} c^{2} x^{4}-588 A \,b^{4} c^{2} x^{2}+308 B \,b^{5} c \,x^{2}+1764 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{5} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-882 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{5} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-924 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{6} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+462 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{6} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right )}{69615 \left (c \,x^{2}+b \right )^{2} c^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} x^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^{3/2}\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \]
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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