Optimal. Leaf size=356 \[ \frac {4 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 A c+b B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}-\frac {8 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 A c+b B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}+\frac {4}{15} \sqrt {x} \sqrt {b x^2+c x^4} (9 A c+b B)+\frac {2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac {8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \]
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Rubi [A] time = 0.45, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2038, 2021, 2032, 329, 305, 220, 1196} \[ \frac {4 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 A c+b B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}-\frac {8 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 A c+b B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}+\frac {2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac {4}{15} \sqrt {x} \sqrt {b x^2+c x^4} (9 A c+b B)+\frac {8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2021
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{9/2}} \, dx &=-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac {\left (2 \left (-\frac {b B}{2}-\frac {9 A c}{2}\right )\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{5/2}} \, dx}{b}\\ &=\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {1}{3} (2 (b B+9 A c)) \int \frac {\sqrt {b x^2+c x^4}}{\sqrt {x}} \, dx\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {1}{15} (4 b (b B+9 A c)) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {\left (4 b (b B+9 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{15 \sqrt {b x^2+c x^4}}\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {\left (8 b (b B+9 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 )}{15 \sqrt {b x^2+c x^4}}\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {\left (8 b^{3/2} (b B+9 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 )}{15 \sqrt {c} \sqrt {b x^2+c x^4}}-\frac {\left (8 b^{3/2} (b B+9 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 )}{15 \sqrt {c} \sqrt {b x^2+c x^4}}\\ &=\frac {8 b (b B+9 A c) x^{3/2} \left (b+c x^2\right )}{15 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac {8 b^{5/4} (b B+9 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 )}{15 c^{3/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{5/4} (b B+9 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 )}{15 c^{3/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 85, normalized size = 0.24 \[ \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (\frac {x^2 (9 A c+b B) \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )}{\sqrt {\frac {c x^2}{b}+1}}-\frac {3 A \left (b+c x^2\right )^2}{b}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{4} + {\left (B b + A c\right )} x^{2} + A b\right )} \sqrt {c x^{4} + b x^{2}}}{x^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 429, normalized size = 1.21 \[ \frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (5 B \,c^{3} x^{6}+9 A \,c^{3} x^{4}+16 B b \,c^{2} x^{4}-36 A b \,c^{2} x^{2}+11 B \,b^{2} c \,x^{2}+108 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{2} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-54 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{2} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+12 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{3} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-45 A \,b^{2} c \right )}{45 \left (c \,x^{2}+b \right )^{2} c \,x^{\frac {7}{2}}} \]
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 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{9/2}} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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