Optimal. Leaf size=150 \[ -\frac {5 a^4 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}-\frac {5 a^3 A x \sqrt {a+c x^2}}{128 c}-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {\left (a+c x^2\right )^{7/2} (16 a B-63 A c x)}{504 c^2}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c} \]
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Rubi [A] time = 0.07, 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 {5 a^4 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}-\frac {5 a^3 A x \sqrt {a+c x^2}}{128 c}-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {\left (a+c x^2\right )^{7/2} (16 a B-63 A c x)}{504 c^2}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{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^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {\int x (-2 a B+9 A c x) \left (a+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {(a A) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 a^2 A\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c}\\ &=-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 a^3 A\right ) \int \sqrt {a+c x^2} \, dx}{64 c}\\ &=-\frac {5 a^3 A x \sqrt {a+c x^2}}{128 c}-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 a^4 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c}\\ &=-\frac {5 a^3 A x \sqrt {a+c x^2}}{128 c}-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 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}\\ &=-\frac {5 a^3 A x \sqrt {a+c x^2}}{128 c}-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {5 a^4 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 131, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-\frac {315 a^{7/2} A \sqrt {c} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}-256 a^4 B+a^3 c x (315 A+128 B x)+6 a^2 c^2 x^3 (413 A+320 B x)+8 a c^3 x^5 (357 A+304 B x)+112 c^4 x^7 (9 A+8 B x)\right )}{8064 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 140, normalized size = 0.93 \begin {gather*} \frac {5 a^4 A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{128 c^{3/2}}+\frac {\sqrt {a+c x^2} \left (-256 a^4 B+315 a^3 A c x+128 a^3 B c x^2+2478 a^2 A c^2 x^3+1920 a^2 B c^2 x^4+2856 a A c^3 x^5+2432 a B c^3 x^6+1008 A c^4 x^7+896 B c^4 x^8\right )}{8064 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 271, normalized size = 1.81 \begin {gather*} \left [\frac {315 \, A a^{4} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (896 \, B c^{4} x^{8} + 1008 \, A c^{4} x^{7} + 2432 \, B a c^{3} x^{6} + 2856 \, A a c^{3} x^{5} + 1920 \, B a^{2} c^{2} x^{4} + 2478 \, A a^{2} c^{2} x^{3} + 128 \, B a^{3} c x^{2} + 315 \, A a^{3} c x - 256 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{16128 \, c^{2}}, \frac {315 \, A a^{4} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (896 \, B c^{4} x^{8} + 1008 \, A c^{4} x^{7} + 2432 \, B a c^{3} x^{6} + 2856 \, A a c^{3} x^{5} + 1920 \, B a^{2} c^{2} x^{4} + 2478 \, A a^{2} c^{2} x^{3} + 128 \, B a^{3} c x^{2} + 315 \, A a^{3} c x - 256 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{8064 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 128, normalized size = 0.85 \begin {gather*} \frac {5 \, A a^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} - \frac {1}{8064} \, {\left (\frac {256 \, B a^{4}}{c^{2}} - {\left (\frac {315 \, A a^{3}}{c} + 2 \, {\left (\frac {64 \, B a^{3}}{c} + {\left (1239 \, A a^{2} + 4 \, {\left (240 \, B a^{2} + {\left (357 \, A a c + 2 \, {\left (152 \, B a c + 7 \, {\left (8 \, B c^{2} x + 9 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 0.88 \begin {gather*} -\frac {5 A \,a^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}-\frac {5 \sqrt {c \,x^{2}+a}\, A \,a^{3} x}{128 c}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,a^{2} x}{192 c}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A a x}{48 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,x^{2}}{9 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A x}{8 c}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B a}{63 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 124, normalized size = 0.83 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B x^{2}}{9 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A a x}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A a^{2} x}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} A a^{3} x}{128 \, c} - \frac {5 \, A a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B a}{63 \, 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^2\,{\left (c\,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.
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sympy [A] time = 24.06, size = 442, normalized size = 2.95 \begin {gather*} \frac {5 A a^{\frac {7}{2}} x}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {133 A a^{\frac {5}{2}} x^{3}}{384 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {127 A a^{\frac {3}{2}} c x^{5}}{192 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {23 A \sqrt {a} c^{2} x^{7}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {5 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {3}{2}}} + \frac {A c^{3} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B a^{2} \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 ) + 2 B 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 ) + B c^{2} \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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