Optimal. Leaf size=175 \[ \frac {3 a^4 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}}+\frac {3 a^3 A x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {a \left (a+c x^2\right )^{5/2} (128 a B-315 A c x)}{5040 c^3}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c} \]
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Rubi [A] time = 0.14, antiderivative size = 175, 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 = {833, 780, 195, 217, 206} \begin {gather*} \frac {3 a^3 A x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac {3 a^4 A \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} (128 a B-315 A c x)}{5040 c^3}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 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^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {\int x^3 (-4 a B+9 A c x) \left (a+c x^2\right )^{3/2} \, dx}{9 c}\\ &=\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {\int x^2 (-27 a A c-32 a B c x) \left (a+c x^2\right )^{3/2} \, dx}{72 c^2}\\ &=-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {\int x \left (64 a^2 B c-189 a A c^2 x\right ) \left (a+c x^2\right )^{3/2} \, dx}{504 c^3}\\ &=-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac {\left (a^2 A\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{16 c^2}\\ &=\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac {\left (3 a^3 A\right ) \int \sqrt {a+c x^2} \, dx}{64 c^2}\\ &=\frac {3 a^3 A x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac {\left (3 a^4 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c^2}\\ &=\frac {3 a^3 A x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac {\left (3 a^4 A\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 A x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac {3 a^4 A \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 = 0.25, size = 132, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+c x^2} \left (\frac {945 a^{7/2} A \sqrt {c} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}+1024 a^4 B-a^3 c x (945 A+512 B x)+6 a^2 c^2 x^3 (105 A+64 B x)+40 a c^3 x^5 (189 A+160 B x)+560 c^4 x^7 (9 A+8 B x)\right )}{40320 c^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 140, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a+c x^2} \left (1024 a^4 B-945 a^3 A c x-512 a^3 B c x^2+630 a^2 A c^2 x^3+384 a^2 B c^2 x^4+7560 a A c^3 x^5+6400 a B c^3 x^6+5040 A c^4 x^7+4480 B c^4 x^8\right )}{40320 c^3}-\frac {3 a^4 A \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 = 0.52, size = 272, normalized size = 1.55 \begin {gather*} \left [\frac {945 \, A a^{4} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (4480 \, B c^{4} x^{8} + 5040 \, A c^{4} x^{7} + 6400 \, B a c^{3} x^{6} + 7560 \, A a c^{3} x^{5} + 384 \, B a^{2} c^{2} x^{4} + 630 \, A a^{2} c^{2} x^{3} - 512 \, B a^{3} c x^{2} - 945 \, A a^{3} c x + 1024 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{80640 \, c^{3}}, -\frac {945 \, A a^{4} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (4480 \, B c^{4} x^{8} + 5040 \, A c^{4} x^{7} + 6400 \, B a c^{3} x^{6} + 7560 \, A a c^{3} x^{5} + 384 \, B a^{2} c^{2} x^{4} + 630 \, A a^{2} c^{2} x^{3} - 512 \, B a^{3} c x^{2} - 945 \, A a^{3} c x + 1024 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{40320 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 130, normalized size = 0.74 \begin {gather*} -\frac {3 \, A a^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {5}{2}}} + \frac {1}{40320} \, \sqrt {c x^{2} + a} {\left (\frac {1024 \, B a^{4}}{c^{3}} - {\left (\frac {945 \, A a^{3}}{c^{2}} + 2 \, {\left (\frac {256 \, B a^{3}}{c^{2}} - {\left (\frac {315 \, A a^{2}}{c} + 4 \, {\left (\frac {48 \, B a^{2}}{c} + 5 \, {\left (189 \, A a + 2 \, {\left (80 \, B a + 7 \, {\left (8 \, B c x + 9 \, A c\right )} x\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 = 155, normalized size = 0.89 \begin {gather*} \frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,x^{4}}{9 c}+\frac {3 A \,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}\, A \,a^{3} x}{128 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,x^{3}}{8 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,a^{2} x}{64 c^{2}}-\frac {4 \left (c \,x^{2}+a \right )^{\frac {5}{2}} B a \,x^{2}}{63 c^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A a x}{16 c^{2}}+\frac {8 \left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,a^{2}}{315 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 147, normalized size = 0.84 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B x^{4}}{9 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A x^{3}}{8 \, c} - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B a x^{2}}{63 \, c^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A a x}{16 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A a^{2} x}{64 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} A a^{3} x}{128 \, c^{2}} + \frac {3 \, A a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {5}{2}}} + \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B a^{2}}{315 \, c^{3}} \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^4\,{\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 = 19.24, size = 366, normalized size = 2.09 \begin {gather*} - \frac {3 A a^{\frac {7}{2}} x}{128 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a^{\frac {5}{2}} x^{3}}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {13 A a^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {5 A \sqrt {a} c x^{7}}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {5}{2}}} + \frac {A c^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B a \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 ) + B c \left (\begin {cases} - \frac {16 a^{4} \sqrt {a + c x^{2}}}{315 c^{4}} + \frac {8 a^{3} x^{2} \sqrt {a + c x^{2}}}{315 c^{3}} - \frac {2 a^{2} x^{4} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{6} \sqrt {a + c x^{2}}}{63 c} + \frac {x^{8} \sqrt {a + c x^{2}}}{9} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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