Optimal. Leaf size=150 \[ \frac {3 a^4 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}}+\frac {3 a^3 B x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac {a \left (a+c x^2\right )^{5/2} (32 A+35 B x)}{560 c^2}+\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c} \]
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Rubi [A] time = 0.09, 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+c x^2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {3 a^4 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}}-\frac {a \left (a+c x^2\right )^{5/2} (32 A+35 B x)}{560 c^2}+\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c} \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+c x^2\right )^{3/2} \, dx &=\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {\int x^2 (-3 a B+8 A c x) \left (a+c x^2\right )^{3/2} \, dx}{8 c}\\ &=\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {\int x (-16 a A c-21 a B c x) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2}\\ &=\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (a^2 B\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{16 c^2}\\ &=\frac {a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (3 a^3 B\right ) \int \sqrt {a+c x^2} \, dx}{64 c^2}\\ &=\frac {3 a^3 B x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (3 a^4 B\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c^2}\\ &=\frac {3 a^3 B x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {\left (3 a^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{128 c^2}\\ &=\frac {3 a^3 B x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac {3 a^4 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 126, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+c x^2} \left (\frac {105 a^{7/2} B \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}+\sqrt {c} \left (-a^3 (256 A+105 B x)+2 a^2 c x^2 (64 A+35 B x)+8 a c^2 x^4 (128 A+105 B x)+80 c^3 x^6 (8 A+7 B x)\right )\right )}{4480 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 125, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-256 a^3 A-105 a^3 B x+128 a^2 A c x^2+70 a^2 B c x^3+1024 a A c^2 x^4+840 a B c^2 x^5+640 A c^3 x^6+560 B c^3 x^7\right )}{4480 c^2}-\frac {3 a^4 B \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{128 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 254, normalized size = 1.69 \begin {gather*} \left [\frac {105 \, B a^{4} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (560 \, B c^{4} x^{7} + 640 \, A c^{4} x^{6} + 840 \, B a c^{3} x^{5} + 1024 \, A a c^{3} x^{4} + 70 \, B a^{2} c^{2} x^{3} + 128 \, A a^{2} c^{2} x^{2} - 105 \, B a^{3} c x - 256 \, A a^{3} c\right )} \sqrt {c x^{2} + a}}{8960 \, c^{3}}, -\frac {105 \, B a^{4} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (560 \, B c^{4} x^{7} + 640 \, A c^{4} x^{6} + 840 \, B a c^{3} x^{5} + 1024 \, A a c^{3} x^{4} + 70 \, B a^{2} c^{2} x^{3} + 128 \, A a^{2} c^{2} x^{2} - 105 \, B a^{3} c x - 256 \, A a^{3} c\right )} \sqrt {c x^{2} + a}}{4480 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 115, normalized size = 0.77 \begin {gather*} -\frac {3 \, B a^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {5}{2}}} - \frac {1}{4480} \, \sqrt {c x^{2} + a} {\left (\frac {256 \, A a^{3}}{c^{2}} + {\left (\frac {105 \, B a^{3}}{c^{2}} - 2 \, {\left (\frac {64 \, A a^{2}}{c} + {\left (\frac {35 \, B a^{2}}{c} + 4 \, {\left (128 \, A a + 5 \, {\left (21 \, B a + 2 \, {\left (7 \, B c x + 8 \, A c\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 = 0.06, size = 134, normalized size = 0.89 \begin {gather*} \frac {3 B \,a^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {5}{2}}}+\frac {3 \sqrt {c \,x^{2}+a}\, B \,a^{3} x}{128 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,x^{3}}{8 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,x^{2}}{7 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{2} x}{64 c^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B a x}{16 c^{2}}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} A a}{35 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 126, normalized size = 0.84 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B x^{3}}{8 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A x^{2}}{7 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B a x}{16 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{64 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} B a^{3} x}{128 \, c^{2}} + \frac {3 \, B a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {5}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A a}{35 \, c^{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 (c\,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 = 18.14, size = 318, normalized size = 2.12 \begin {gather*} A a \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + A c \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - \frac {3 B a^{\frac {7}{2}} x}{128 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {B a^{\frac {5}{2}} x^{3}}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {13 B a^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {5 B \sqrt {a} c x^{7}}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {5}{2}}} + \frac {B c^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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