Optimal. Leaf size=340 \[ -\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 c^{11/4} \sqrt {b x^2+c x^4}}+\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 c^{11/4} \sqrt {b x^2+c x^4}}-\frac {3 x^{3/2} \left (b+c x^2\right ) (7 b B-5 A c)}{5 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {\sqrt {x} \sqrt {b x^2+c x^4} (7 b B-5 A c)}{5 b c^2}-\frac {x^{9/2} (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2037, 2024, 2032, 329, 305, 220, 1196} \[ -\frac {3 x^{3/2} \left (b+c x^2\right ) (7 b B-5 A c)}{5 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {\sqrt {x} \sqrt {b x^2+c x^4} (7 b B-5 A c)}{5 b c^2}-\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 c^{11/4} \sqrt {b x^2+c x^4}}+\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 c^{11/4} \sqrt {b x^2+c x^4}}-\frac {x^{9/2} (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2024
Rule 2032
Rule 2037
Rubi steps
\begin {align*} \int \frac {x^{11/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {(b B-A c) x^{9/2}}{b c \sqrt {b x^2+c x^4}}+\frac {\left (\frac {7 b B}{2}-\frac {5 A c}{2}\right ) \int \frac {x^{7/2}}{\sqrt {b x^2+c x^4}} \, dx}{b c}\\ &=-\frac {(b B-A c) x^{9/2}}{b c \sqrt {b x^2+c x^4}}+\frac {(7 b B-5 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{5 b c^2}-\frac {(3 (7 b B-5 A c)) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{10 c^2}\\ &=-\frac {(b B-A c) x^{9/2}}{b c \sqrt {b x^2+c x^4}}+\frac {(7 b B-5 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{5 b c^2}-\frac {\left (3 (7 b B-5 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{10 c^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{9/2}}{b c \sqrt {b x^2+c x^4}}+\frac {(7 b B-5 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{5 b c^2}-\frac {\left (3 (7 b B-5 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 )}{5 c^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{9/2}}{b c \sqrt {b x^2+c x^4}}+\frac {(7 b B-5 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{5 b c^2}-\frac {\left (3 \sqrt {b} (7 b B-5 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 )}{5 c^{5/2} \sqrt {b x^2+c x^4}}+\frac {\left (3 \sqrt {b} (7 b B-5 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 )}{5 c^{5/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{9/2}}{b c \sqrt {b x^2+c x^4}}-\frac {3 (7 b B-5 A c) x^{3/2} \left (b+c x^2\right )}{5 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {(7 b B-5 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{5 b c^2}+\frac {3 \sqrt [4]{b} (7 b B-5 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 )}{5 c^{11/4} \sqrt {b x^2+c x^4}}-\frac {3 \sqrt [4]{b} (7 b B-5 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 )}{10 c^{11/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.13, size = 85, normalized size = 0.25 \[ \frac {2 x^{5/2} \left (\sqrt {\frac {c x^2}{b}+1} (7 b B-5 A c) \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {c x^2}{b}\right )+5 A c-7 b B+B c x^2\right )}{5 c^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{3} + A x\right )} \sqrt {x}}{c^{2} x^{4} + 2 \, b c x^{2} + b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {11}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 394, normalized size = 1.16 \[ \frac {\left (c \,x^{2}+b \right ) \left (4 B \,c^{2} x^{4}-10 A \,c^{2} x^{2}+14 B b c \,x^{2}+30 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A b c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-15 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A b c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-42 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) x^{\frac {5}{2}}}{10 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {11}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{11/2}\,\left (B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________