Optimal. Leaf size=201 \[ -\frac {4 b^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-11 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{5/4} \sqrt {b x^2+c x^4}}-\frac {4 b \sqrt {b x^2+c x^4} (b B-11 A c)}{77 c \sqrt {x}}-\frac {2 \left (b x^2+c x^4\right )^{3/2} (b B-11 A c)}{77 c x^{5/2}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2039, 2021, 2032, 329, 220} \[ -\frac {4 b^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-11 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{5/4} \sqrt {b x^2+c x^4}}-\frac {2 \left (b x^2+c x^4\right )^{3/2} (b B-11 A c)}{77 c x^{5/2}}-\frac {4 b \sqrt {b x^2+c x^4} (b B-11 A c)}{77 c \sqrt {x}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 329
Rule 2021
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{7/2}} \, dx &=\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}}-\frac {\left (2 \left (\frac {b B}{2}-\frac {11 A c}{2}\right )\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{7/2}} \, dx}{11 c}\\ &=-\frac {2 (b B-11 A c) \left (b x^2+c x^4\right )^{3/2}}{77 c x^{5/2}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}}-\frac {(6 b (b B-11 A c)) \int \frac {\sqrt {b x^2+c x^4}}{x^{3/2}} \, dx}{77 c}\\ &=-\frac {4 b (b B-11 A c) \sqrt {b x^2+c x^4}}{77 c \sqrt {x}}-\frac {2 (b B-11 A c) \left (b x^2+c x^4\right )^{3/2}}{77 c x^{5/2}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}}-\frac {\left (4 b^2 (b B-11 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{77 c}\\ &=-\frac {4 b (b B-11 A c) \sqrt {b x^2+c x^4}}{77 c \sqrt {x}}-\frac {2 (b B-11 A c) \left (b x^2+c x^4\right )^{3/2}}{77 c x^{5/2}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}}-\frac {\left (4 b^2 (b B-11 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{77 c \sqrt {b x^2+c x^4}}\\ &=-\frac {4 b (b B-11 A c) \sqrt {b x^2+c x^4}}{77 c \sqrt {x}}-\frac {2 (b B-11 A c) \left (b x^2+c x^4\right )^{3/2}}{77 c x^{5/2}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}}-\frac {\left (8 b^2 (b B-11 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 )}{77 c \sqrt {b x^2+c x^4}}\\ &=-\frac {4 b (b B-11 A c) \sqrt {b x^2+c x^4}}{77 c \sqrt {x}}-\frac {2 (b B-11 A c) \left (b x^2+c x^4\right )^{3/2}}{77 c x^{5/2}}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{11 c x^{9/2}}-\frac {4 b^{7/4} (b B-11 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 )}{77 c^{5/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 97, normalized size = 0.48 \[ \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (b (11 A c-b B) \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{b}\right )+B \sqrt {\frac {c x^2}{b}+1} \left (b+c x^2\right )^2\right )}{11 c \sqrt {x} \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{4} + {\left (B b + A c\right )} x^{2} + A b\right )} \sqrt {c x^{4} + b x^{2}}}{x^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 283, normalized size = 1.41 \[ \frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (7 B \,c^{4} x^{7}+11 A \,c^{4} x^{5}+20 B b \,c^{3} x^{5}+44 A b \,c^{3} x^{3}+17 B \,b^{2} c^{2} x^{3}+33 A \,b^{2} c^{2} x +4 B \,b^{3} c x +22 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{2} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right )}{77 \left (c \,x^{2}+b \right )^{2} c^{2} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________