Optimal. Leaf size=150 \[ \frac {3 a^4 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}+\frac {3 a^3 B x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}-\frac {a \left (a+b x^2\right )^{5/2} (32 A+35 B x)}{560 b^2}+\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]
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Rubi [A] time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {833, 780, 195, 217, 206} \begin {gather*} \frac {3 a^3 B x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {3 a^4 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}-\frac {a \left (a+b x^2\right )^{5/2} (32 A+35 B x)}{560 b^2}+\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 780
Rule 833
Rubi steps
\begin {align*} \int x^3 (A+B x) \left (a+b x^2\right )^{3/2} \, dx &=\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {\int x^2 (-3 a B+8 A b x) \left (a+b x^2\right )^{3/2} \, dx}{8 b}\\ &=\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {\int x (-16 a A b-21 a b B x) \left (a+b x^2\right )^{3/2} \, dx}{56 b^2}\\ &=\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a (32 A+35 B x) \left (a+b x^2\right )^{5/2}}{560 b^2}+\frac {\left (a^2 B\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{16 b^2}\\ &=\frac {a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a (32 A+35 B x) \left (a+b x^2\right )^{5/2}}{560 b^2}+\frac {\left (3 a^3 B\right ) \int \sqrt {a+b x^2} \, dx}{64 b^2}\\ &=\frac {3 a^3 B x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a (32 A+35 B x) \left (a+b x^2\right )^{5/2}}{560 b^2}+\frac {\left (3 a^4 B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^2}\\ &=\frac {3 a^3 B x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a (32 A+35 B x) \left (a+b x^2\right )^{5/2}}{560 b^2}+\frac {\left (3 a^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^2}\\ &=\frac {3 a^3 B x \sqrt {a+b x^2}}{128 b^2}+\frac {a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac {B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {a (32 A+35 B x) \left (a+b x^2\right )^{5/2}}{560 b^2}+\frac {3 a^4 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 126, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+b x^2} \left (\frac {105 a^{7/2} B \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}+\sqrt {b} \left (-a^3 (256 A+105 B x)+2 a^2 b x^2 (64 A+35 B x)+8 a b^2 x^4 (128 A+105 B x)+80 b^3 x^6 (8 A+7 B x)\right )\right )}{4480 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 125, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-256 a^3 A-105 a^3 B x+128 a^2 A b x^2+70 a^2 b B x^3+1024 a A b^2 x^4+840 a b^2 B x^5+640 A b^3 x^6+560 b^3 B x^7\right )}{4480 b^2}-\frac {3 a^4 B \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{128 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 254, normalized size = 1.69 \begin {gather*} \left [\frac {105 \, B a^{4} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (560 \, B b^{4} x^{7} + 640 \, A b^{4} x^{6} + 840 \, B a b^{3} x^{5} + 1024 \, A a b^{3} x^{4} + 70 \, B a^{2} b^{2} x^{3} + 128 \, A a^{2} b^{2} x^{2} - 105 \, B a^{3} b x - 256 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{8960 \, b^{3}}, -\frac {105 \, B a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (560 \, B b^{4} x^{7} + 640 \, A b^{4} x^{6} + 840 \, B a b^{3} x^{5} + 1024 \, A a b^{3} x^{4} + 70 \, B a^{2} b^{2} x^{3} + 128 \, A a^{2} b^{2} x^{2} - 105 \, B a^{3} b x - 256 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{4480 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 115, normalized size = 0.77 \begin {gather*} -\frac {3 \, B a^{4} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} - \frac {1}{4480} \, \sqrt {b x^{2} + a} {\left (\frac {256 \, A a^{3}}{b^{2}} + {\left (\frac {105 \, B a^{3}}{b^{2}} - 2 \, {\left (\frac {64 \, A a^{2}}{b} + {\left (\frac {35 \, B a^{2}}{b} + 4 \, {\left (128 \, A a + 5 \, {\left (21 \, B a + 2 \, {\left (7 \, B b x + 8 \, A b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 0.89 \begin {gather*} \frac {3 B \,a^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}+\frac {3 \sqrt {b \,x^{2}+a}\, B \,a^{3} x}{128 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B \,x^{3}}{8 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,x^{2}}{7 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{2} x}{64 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B a x}{16 b^{2}}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {5}{2}} A a}{35 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 126, normalized size = 0.84 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{3}}{8 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x^{2}}{7 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{3} x}{128 \, b^{2}} + \frac {3 \, B a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a}{35 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.79, size = 318, normalized size = 2.12 \begin {gather*} A a \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 ) + 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 ) - \frac {3 B a^{\frac {7}{2}} x}{128 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {5}{2}} x^{3}}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {13 B a^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 B \sqrt {a} b x^{7}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} + \frac {B b^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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