Optimal. Leaf size=152 \[ -\frac {2 b^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+b x^2}}+\frac {2 \sqrt {a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {453, 277, 329, 220} \[ -\frac {2 b^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+b x^2}}+\frac {2 \sqrt {a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 277
Rule 329
Rule 453
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx &=-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {\left (2 \left (\frac {A b}{2}-\frac {7 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x^2}}{x^{5/2}} \, dx}{7 a}\\ &=\frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {(2 b (A b-7 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{21 a}\\ &=\frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {(4 b (A b-7 a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{21 a}\\ &=\frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {2 b^{3/4} (A b-7 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.09, size = 79, normalized size = 0.52 \[ \frac {2 \sqrt {a+b x^2} \left (\frac {x^2 (A b-7 a B) \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{4};-\frac {b x^2}{a}\right )}{\sqrt {\frac {b x^2}{a}+1}}-3 A \left (a+b x^2\right )\right )}{21 a x^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {9}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 242, normalized size = 1.59 \[ -\frac {2 \left (2 A \,b^{2} x^{4}+7 B a b \,x^{4}+\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, A b \,x^{3} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-7 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, B a \,x^{3} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+5 A a b \,x^{2}+7 B \,a^{2} x^{2}+3 A \,a^{2}\right )}{21 \sqrt {b \,x^{2}+a}\, a \,x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (B\,x^2+A\right )\,\sqrt {b\,x^2+a}}{x^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 21.30, size = 97, normalized size = 0.64 \[ \frac {A \sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________