Optimal. Leaf size=87 \[ \frac {3 a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{4} A x \left (a+c x^2\right )^{3/2}+\frac {3}{8} a A x \sqrt {a+c x^2}+\frac {B \left (a+c x^2\right )^{5/2}}{5 c} \]
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Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \begin {gather*} \frac {3 a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{4} A x \left (a+c x^2\right )^{3/2}+\frac {3}{8} a A x \sqrt {a+c x^2}+\frac {B \left (a+c x^2\right )^{5/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rubi steps
\begin {align*} \int (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac {B \left (a+c x^2\right )^{5/2}}{5 c}+A \int \left (a+c x^2\right )^{3/2} \, dx\\ &=\frac {1}{4} A x \left (a+c x^2\right )^{3/2}+\frac {B \left (a+c x^2\right )^{5/2}}{5 c}+\frac {1}{4} (3 a A) \int \sqrt {a+c x^2} \, dx\\ &=\frac {3}{8} a A x \sqrt {a+c x^2}+\frac {1}{4} A x \left (a+c x^2\right )^{3/2}+\frac {B \left (a+c x^2\right )^{5/2}}{5 c}+\frac {1}{8} \left (3 a^2 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {3}{8} a A x \sqrt {a+c x^2}+\frac {1}{4} A x \left (a+c x^2\right )^{3/2}+\frac {B \left (a+c x^2\right )^{5/2}}{5 c}+\frac {1}{8} \left (3 a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {3}{8} a A x \sqrt {a+c x^2}+\frac {1}{4} A x \left (a+c x^2\right )^{3/2}+\frac {B \left (a+c x^2\right )^{5/2}}{5 c}+\frac {3 a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 1.01 \begin {gather*} \frac {\sqrt {a+c x^2} \left (8 a^2 B+a c x (25 A+16 B x)+2 c^2 x^3 (5 A+4 B x)\right )+15 a^2 A \sqrt {c} \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{40 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 92, normalized size = 1.06 \begin {gather*} \frac {\sqrt {a+c x^2} \left (8 a^2 B+25 a A c x+16 a B c x^2+10 A c^2 x^3+8 B c^2 x^4\right )}{40 c}-\frac {3 a^2 A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 176, normalized size = 2.02 \begin {gather*} \left [\frac {15 \, A a^{2} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{80 \, c}, -\frac {15 \, A a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{40 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 76, normalized size = 0.87 \begin {gather*} -\frac {3 \, A a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, \sqrt {c}} + \frac {1}{40} \, \sqrt {c x^{2} + a} {\left (\frac {8 \, B a^{2}}{c} + {\left (25 \, A a + 2 \, {\left (8 \, B a + {\left (4 \, B c x + 5 \, A c\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 = 0.05, size = 69, normalized size = 0.79 \begin {gather*} \frac {3 A \,a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}+\frac {3 \sqrt {c \,x^{2}+a}\, A a x}{8}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A x}{4}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 61, normalized size = 0.70 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A x + \frac {3}{8} \, \sqrt {c x^{2} + a} A a x + \frac {3 \, A a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B}{5 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 54, normalized size = 0.62 \begin {gather*} \frac {B\,{\left (c\,x^2+a\right )}^{5/2}}{5\,c}+\frac {A\,x\,{\left (c\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.62, size = 219, normalized size = 2.52 \begin {gather*} \frac {A a^{\frac {3}{2}} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {A a^{\frac {3}{2}} x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A \sqrt {a} c x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + \frac {A c^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B a \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + B c \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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