Optimal. Leaf size=173 \[ \frac {3 a^4 B x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {3 a^5 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {847, 794, 201,
223, 212} \begin {gather*} \frac {3 a^5 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}+\frac {3 a^4 B x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}-\frac {a \left (a+b x^2\right )^{7/2} (160 A+189 B x)}{5040 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 212
Rule 223
Rule 794
Rule 847
Rubi steps
\begin {align*} \int x^3 (A+B x) \left (a+b x^2\right )^{5/2} \, dx &=\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {\int x^2 (-3 a B+10 A b x) \left (a+b x^2\right )^{5/2} \, dx}{10 b}\\ &=\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {\int x (-20 a A b-27 a b B x) \left (a+b x^2\right )^{5/2} \, dx}{90 b^2}\\ &=\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {\left (3 a^2 B\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{80 b^2}\\ &=\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {\left (a^3 B\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{32 b^2}\\ &=\frac {a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {\left (3 a^4 B\right ) \int \sqrt {a+b x^2} \, dx}{128 b^2}\\ &=\frac {3 a^4 B x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {\left (3 a^5 B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^2}\\ &=\frac {3 a^4 B x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {\left (3 a^5 B\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^2}\\ &=\frac {3 a^4 B x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac {B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {3 a^5 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.33, size = 137, normalized size = 0.79 \begin {gather*} \frac {\sqrt {b} \sqrt {a+b x^2} \left (896 b^4 x^8 (10 A+9 B x)+10 a^3 b x^2 (128 A+63 B x)-5 a^4 (512 A+189 B x)+24 a^2 b^2 x^4 (800 A+651 B x)+16 a b^3 x^6 (1520 A+1323 B x)\right )-945 a^5 B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{80640 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 152, normalized size = 0.88
method | result | size |
risch | \(-\frac {\left (-8064 B \,b^{4} x^{9}-8960 A \,b^{4} x^{8}-21168 B a \,b^{3} x^{7}-24320 a \,b^{3} A \,x^{6}-15624 B \,a^{2} b^{2} x^{5}-19200 a^{2} A \,b^{2} x^{4}-630 B \,a^{3} b \,x^{3}-1280 A \,a^{3} b \,x^{2}+945 B \,a^{4} x +2560 a^{4} A \right ) \sqrt {b \,x^{2}+a}}{80640 b^{2}}+\frac {3 a^{5} B \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}\) | \(137\) |
default | \(B \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+A \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 145, normalized size = 0.84 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{3}}{10 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x^{2}}{9 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{4} x}{256 \, b^{2}} + \frac {3 \, B a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a}{63 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.26, size = 302, normalized size = 1.75 \begin {gather*} \left [\frac {945 \, B a^{5} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (8064 \, B b^{5} x^{9} + 8960 \, A b^{5} x^{8} + 21168 \, B a b^{4} x^{7} + 24320 \, A a b^{4} x^{6} + 15624 \, B a^{2} b^{3} x^{5} + 19200 \, A a^{2} b^{3} x^{4} + 630 \, B a^{3} b^{2} x^{3} + 1280 \, A a^{3} b^{2} x^{2} - 945 \, B a^{4} b x - 2560 \, A a^{4} b\right )} \sqrt {b x^{2} + a}}{161280 \, b^{3}}, -\frac {945 \, B a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8064 \, B b^{5} x^{9} + 8960 \, A b^{5} x^{8} + 21168 \, B a b^{4} x^{7} + 24320 \, A a b^{4} x^{6} + 15624 \, B a^{2} b^{3} x^{5} + 19200 \, A a^{2} b^{3} x^{4} + 630 \, B a^{3} b^{2} x^{3} + 1280 \, A a^{3} b^{2} x^{2} - 945 \, B a^{4} b x - 2560 \, A a^{4} b\right )} \sqrt {b x^{2} + a}}{80640 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 136.14, size = 469, normalized size = 2.71 \begin {gather*} A a^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + b x^{2}}}{15 b^{2}} + \frac {a x^{2} \sqrt {a + b x^{2}}}{15 b} + \frac {x^{4} \sqrt {a + b x^{2}}}{5} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 A a b \left (\begin {cases} \frac {8 a^{3} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + b x^{2}}}{105 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{2}}}{35 b} + \frac {x^{6} \sqrt {a + b x^{2}}}{7} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + A b^{2} \left (\begin {cases} - \frac {16 a^{4} \sqrt {a + b x^{2}}}{315 b^{4}} + \frac {8 a^{3} x^{2} \sqrt {a + b x^{2}}}{315 b^{3}} - \frac {2 a^{2} x^{4} \sqrt {a + b x^{2}}}{105 b^{2}} + \frac {a x^{6} \sqrt {a + b x^{2}}}{63 b} + \frac {x^{8} \sqrt {a + b x^{2}}}{9} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) - \frac {3 B a^{\frac {9}{2}} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {7}{2}} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {129 B a^{\frac {5}{2}} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {73 B a^{\frac {3}{2}} b x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 B \sqrt {a} b^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} + \frac {B b^{3} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.47, size = 140, normalized size = 0.81 \begin {gather*} -\frac {3 \, B a^{5} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} - \frac {1}{80640} \, {\left (\frac {2560 \, A a^{4}}{b^{2}} + {\left (\frac {945 \, B a^{4}}{b^{2}} - 2 \, {\left (\frac {640 \, A a^{3}}{b} + {\left (\frac {315 \, B a^{3}}{b} + 4 \, {\left (2400 \, A a^{2} + {\left (1953 \, B a^{2} + 2 \, {\left (1520 \, A a b + 7 \, {\left (189 \, B a b + 8 \, {\left (9 \, B b^{2} x + 10 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {b x^{2} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________