Optimal. Leaf size=152 \[ \frac {5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}}-\frac {5 a x \sqrt {a+b x^2} (6 A b-7 a B)}{16 b^4}+\frac {5 x^3 \sqrt {a+b x^2} (6 A b-7 a B)}{24 b^3}-\frac {x^5 (6 A b-7 a B)}{6 b^2 \sqrt {a+b x^2}}+\frac {B x^7}{6 b \sqrt {a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {459, 288, 321, 217, 206} \[ \frac {5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}}-\frac {x^5 (6 A b-7 a B)}{6 b^2 \sqrt {a+b x^2}}+\frac {5 x^3 \sqrt {a+b x^2} (6 A b-7 a B)}{24 b^3}-\frac {5 a x \sqrt {a+b x^2} (6 A b-7 a B)}{16 b^4}+\frac {B x^7}{6 b \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 288
Rule 321
Rule 459
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {B x^7}{6 b \sqrt {a+b x^2}}-\frac {(-6 A b+7 a B) \int \frac {x^6}{\left (a+b x^2\right )^{3/2}} \, dx}{6 b}\\ &=-\frac {(6 A b-7 a B) x^5}{6 b^2 \sqrt {a+b x^2}}+\frac {B x^7}{6 b \sqrt {a+b x^2}}+\frac {(5 (6 A b-7 a B)) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{6 b^2}\\ &=-\frac {(6 A b-7 a B) x^5}{6 b^2 \sqrt {a+b x^2}}+\frac {B x^7}{6 b \sqrt {a+b x^2}}+\frac {5 (6 A b-7 a B) x^3 \sqrt {a+b x^2}}{24 b^3}-\frac {(5 a (6 A b-7 a B)) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{8 b^3}\\ &=-\frac {(6 A b-7 a B) x^5}{6 b^2 \sqrt {a+b x^2}}+\frac {B x^7}{6 b \sqrt {a+b x^2}}-\frac {5 a (6 A b-7 a B) x \sqrt {a+b x^2}}{16 b^4}+\frac {5 (6 A b-7 a B) x^3 \sqrt {a+b x^2}}{24 b^3}+\frac {\left (5 a^2 (6 A b-7 a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b^4}\\ &=-\frac {(6 A b-7 a B) x^5}{6 b^2 \sqrt {a+b x^2}}+\frac {B x^7}{6 b \sqrt {a+b x^2}}-\frac {5 a (6 A b-7 a B) x \sqrt {a+b x^2}}{16 b^4}+\frac {5 (6 A b-7 a B) x^3 \sqrt {a+b x^2}}{24 b^3}+\frac {\left (5 a^2 (6 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^4}\\ &=-\frac {(6 A b-7 a B) x^5}{6 b^2 \sqrt {a+b x^2}}+\frac {B x^7}{6 b \sqrt {a+b x^2}}-\frac {5 a (6 A b-7 a B) x \sqrt {a+b x^2}}{16 b^4}+\frac {5 (6 A b-7 a B) x^3 \sqrt {a+b x^2}}{24 b^3}+\frac {5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 131, normalized size = 0.86 \[ \frac {\sqrt {b} x \left (105 a^3 B+a^2 \left (35 b B x^2-90 A b\right )-2 a b^2 x^2 \left (15 A+7 B x^2\right )+4 b^3 x^4 \left (3 A+2 B x^2\right )\right )-15 a^{5/2} \sqrt {\frac {b x^2}{a}+1} (7 a B-6 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48 b^{9/2} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.97, size = 325, normalized size = 2.14 \[ \left [-\frac {15 \, {\left (7 \, B a^{4} - 6 \, A a^{3} b + {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, B b^{4} x^{7} - 2 \, {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x^{3} + 15 \, {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {15 \, {\left (7 \, B a^{4} - 6 \, A a^{3} b + {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (8 \, B b^{4} x^{7} - 2 \, {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x^{3} + 15 \, {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 136, normalized size = 0.89 \[ \frac {{\left ({\left (2 \, {\left (\frac {4 \, B x^{2}}{b} - \frac {7 \, B a b^{5} - 6 \, A b^{6}}{b^{7}}\right )} x^{2} + \frac {5 \, {\left (7 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )}}{b^{7}}\right )} x^{2} + \frac {15 \, {\left (7 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )}}{b^{7}}\right )} x}{48 \, \sqrt {b x^{2} + a}} + \frac {5 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 185, normalized size = 1.22 \[ \frac {B \,x^{7}}{6 \sqrt {b \,x^{2}+a}\, b}+\frac {A \,x^{5}}{4 \sqrt {b \,x^{2}+a}\, b}-\frac {7 B a \,x^{5}}{24 \sqrt {b \,x^{2}+a}\, b^{2}}-\frac {5 A a \,x^{3}}{8 \sqrt {b \,x^{2}+a}\, b^{2}}+\frac {35 B \,a^{2} x^{3}}{48 \sqrt {b \,x^{2}+a}\, b^{3}}-\frac {15 A \,a^{2} x}{8 \sqrt {b \,x^{2}+a}\, b^{3}}+\frac {35 B \,a^{3} x}{16 \sqrt {b \,x^{2}+a}\, b^{4}}+\frac {15 A \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {7}{2}}}-\frac {35 B \,a^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.06, size = 170, normalized size = 1.12 \[ \frac {B x^{7}}{6 \, \sqrt {b x^{2} + a} b} - \frac {7 \, B a x^{5}}{24 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A x^{5}}{4 \, \sqrt {b x^{2} + a} b} + \frac {35 \, B a^{2} x^{3}}{48 \, \sqrt {b x^{2} + a} b^{3}} - \frac {5 \, A a x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {35 \, B a^{3} x}{16 \, \sqrt {b x^{2} + a} b^{4}} - \frac {15 \, A a^{2} x}{8 \, \sqrt {b x^{2} + a} b^{3}} - \frac {35 \, B a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {9}{2}}} + \frac {15 \, A a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^6\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 35.61, size = 233, normalized size = 1.53 \[ A \left (- \frac {15 a^{\frac {3}{2}} x}{8 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 \sqrt {a} x^{3}}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{5}}{4 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\frac {35 a^{\frac {5}{2}} x}{16 b^{4} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} x^{3}}{48 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {7 \sqrt {a} x^{5}}{24 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {35 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {9}{2}}} + \frac {x^{7}}{6 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________