Optimal. Leaf size=107 \[ \frac {5 a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}}+\frac {5}{16} a^2 A x \sqrt {a+c x^2}+\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c} \]
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Rubi [A] time = 0.03, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \[ \frac {5}{16} a^2 A x \sqrt {a+c x^2}+\frac {5 a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}}+\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c} \]
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 )^{5/2} \, dx &=\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+A \int \left (a+c x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{6} (5 a A) \int \left (a+c x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{8} \left (5 a^2 A\right ) \int \sqrt {a+c x^2} \, dx\\ &=\frac {5}{16} a^2 A x \sqrt {a+c x^2}+\frac {5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{16} \left (5 a^3 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {5}{16} a^2 A x \sqrt {a+c x^2}+\frac {5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{16} \left (5 a^3 A\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {5}{16} a^2 A x \sqrt {a+c x^2}+\frac {5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac {1}{6} A x \left (a+c x^2\right )^{5/2}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+\frac {5 a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 108, normalized size = 1.01 \[ \frac {105 a^3 A \sqrt {c} \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )+\sqrt {a+c x^2} \left (48 a^3 B+3 a^2 c x (77 A+48 B x)+2 a c^2 x^3 (91 A+72 B x)+8 c^3 x^5 (7 A+6 B x)\right )}{336 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 224, normalized size = 2.09 \[ \left [\frac {105 \, A a^{3} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (48 \, B c^{3} x^{6} + 56 \, A c^{3} x^{5} + 144 \, B a c^{2} x^{4} + 182 \, A a c^{2} x^{3} + 144 \, B a^{2} c x^{2} + 231 \, A a^{2} c x + 48 \, B a^{3}\right )} \sqrt {c x^{2} + a}}{672 \, c}, -\frac {105 \, A a^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (48 \, B c^{3} x^{6} + 56 \, A c^{3} x^{5} + 144 \, B a c^{2} x^{4} + 182 \, A a c^{2} x^{3} + 144 \, B a^{2} c x^{2} + 231 \, A a^{2} c x + 48 \, B a^{3}\right )} \sqrt {c x^{2} + a}}{336 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 101, normalized size = 0.94 \[ -\frac {5 \, A a^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, \sqrt {c}} + \frac {1}{336} \, {\left (\frac {48 \, B a^{3}}{c} + {\left (231 \, A a^{2} + 2 \, {\left (72 \, B a^{2} + {\left (91 \, A a c + 4 \, {\left (18 \, B a c + {\left (6 \, B c^{2} x + 7 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 85, normalized size = 0.79 \[ \frac {5 A \,a^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 \sqrt {c}}+\frac {5 \sqrt {c \,x^{2}+a}\, A \,a^{2} x}{16}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A a x}{24}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A x}{6}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{7 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 77, normalized size = 0.72 \[ \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A a x + \frac {5}{16} \, \sqrt {c x^{2} + a} A a^{2} x + \frac {5 \, A a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{7 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 54, normalized size = 0.50 \[ \frac {B\,{\left (c\,x^2+a\right )}^{7/2}}{7\,c}+\frac {A\,x\,{\left (c\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.70, size = 348, normalized size = 3.25 \[ \frac {A a^{\frac {5}{2}} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 A a^{\frac {5}{2}} x}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 A a^{\frac {3}{2}} c x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 A \sqrt {a} c^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {5 A a^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 \sqrt {c}} + \frac {A c^{3} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B a^{2} \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + 2 B a c \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + B c^{2} \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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