Optimal. Leaf size=103 \[ \frac {a^2 (A b-a B) \left (a+b x^2\right )^{3/2}}{3 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78}
\begin {gather*} \frac {a^2 \left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^4}+\frac {\left (a+b x^2\right )^{7/2} (A b-3 a B)}{7 b^4}-\frac {a \left (a+b x^2\right )^{5/2} (2 A b-3 a B)}{5 b^4}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 457
Rubi steps
\begin {align*} \int x^5 \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 \sqrt {a+b x} (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B) \sqrt {a+b x}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{3/2}}{b^3}+\frac {(A b-3 a B) (a+b x)^{5/2}}{b^3}+\frac {B (a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 (A b-a B) \left (a+b x^2\right )^{3/2}}{3 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 75, normalized size = 0.73 \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} \left (-16 a^3 B+24 a^2 b \left (A+B x^2\right )-6 a b^2 x^2 \left (6 A+5 B x^2\right )+5 b^3 x^4 \left (9 A+7 B x^2\right )\right )}{315 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 144, normalized size = 1.40
method | result | size |
gosper | \(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (35 B \,x^{6} b^{3}+45 A \,b^{3} x^{4}-30 B a \,b^{2} x^{4}-36 A a \,b^{2} x^{2}+24 B \,a^{2} b \,x^{2}+24 A \,a^{2} b -16 B \,a^{3}\right )}{315 b^{4}}\) | \(77\) |
trager | \(\frac {\left (35 B \,b^{4} x^{8}+45 A \,b^{4} x^{6}+5 B a \,b^{3} x^{6}+9 A a \,b^{3} x^{4}-6 B \,a^{2} b^{2} x^{4}-12 a^{2} A \,b^{2} x^{2}+8 B \,a^{3} b \,x^{2}+24 A \,a^{3} b -16 B \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{315 b^{4}}\) | \(101\) |
risch | \(\frac {\left (35 B \,b^{4} x^{8}+45 A \,b^{4} x^{6}+5 B a \,b^{3} x^{6}+9 A a \,b^{3} x^{4}-6 B \,a^{2} b^{2} x^{4}-12 a^{2} A \,b^{2} x^{2}+8 B \,a^{3} b \,x^{2}+24 A \,a^{3} b -16 B \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{315 b^{4}}\) | \(101\) |
default | \(B \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 b}-\frac {2 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 b^{2}}\right )}{7 b}\right )}{3 b}\right )+A \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 b^{2}}\right )}{7 b}\right )\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 132, normalized size = 1.28 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B x^{6}}{9 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a x^{4}}{21 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A x^{4}}{7 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x^{2}}{105 \, b^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a x^{2}}{35 \, b^{2}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3}}{315 \, b^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2}}{105 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.27, size = 99, normalized size = 0.96 \begin {gather*} \frac {{\left (35 \, B b^{4} x^{8} + 5 \, {\left (B a b^{3} + 9 \, A b^{4}\right )} x^{6} - 16 \, B a^{4} + 24 \, A a^{3} b - 3 \, {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + 4 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs.
\(2 (94) = 188\).
time = 0.23, size = 212, normalized size = 2.06 \begin {gather*} \begin {cases} \frac {8 A a^{3} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {4 A a^{2} x^{2} \sqrt {a + b x^{2}}}{105 b^{2}} + \frac {A a x^{4} \sqrt {a + b x^{2}}}{35 b} + \frac {A x^{6} \sqrt {a + b x^{2}}}{7} - \frac {16 B a^{4} \sqrt {a + b x^{2}}}{315 b^{4}} + \frac {8 B a^{3} x^{2} \sqrt {a + b x^{2}}}{315 b^{3}} - \frac {2 B a^{2} x^{4} \sqrt {a + b x^{2}}}{105 b^{2}} + \frac {B a x^{6} \sqrt {a + b x^{2}}}{63 b} + \frac {B x^{8} \sqrt {a + b x^{2}}}{9} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {A x^{6}}{6} + \frac {B x^{8}}{8}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.44, size = 104, normalized size = 1.01 \begin {gather*} \frac {35 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B - 135 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a + 189 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} - 105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} + 45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b - 126 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a b + 105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} b}{315 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.30, size = 96, normalized size = 0.93 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {B\,x^8}{9}-\frac {16\,B\,a^4-24\,A\,a^3\,b}{315\,b^4}+\frac {x^6\,\left (45\,A\,b^4+5\,B\,a\,b^3\right )}{315\,b^4}-\frac {4\,a^2\,x^2\,\left (3\,A\,b-2\,B\,a\right )}{315\,b^3}+\frac {a\,x^4\,\left (3\,A\,b-2\,B\,a\right )}{105\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________