Optimal. Leaf size=130 \[ \frac {5}{8} a^2 x \sqrt {c x^2-\frac {a^2 c}{b^2}}+\frac {5 a b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}-\frac {5 a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}\right )}{8 b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {671, 641, 195, 217, 206} \begin {gather*} \frac {5}{8} a^2 x \sqrt {c x^2-\frac {a^2 c}{b^2}}+\frac {5 a b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}-\frac {5 a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}\right )}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rule 671
Rubi steps
\begin {align*} \int (a+b x)^2 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx &=\frac {b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{4 c}+\frac {1}{4} (5 a) \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx\\ &=\frac {5 a b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{4 c}+\frac {1}{4} \left (5 a^2\right ) \int \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx\\ &=\frac {5}{8} a^2 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {5 a b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{4 c}-\frac {\left (5 a^4 c\right ) \int \frac {1}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}} \, dx}{8 b^2}\\ &=\frac {5}{8} a^2 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {5 a b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{4 c}-\frac {\left (5 a^4 c\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{8 b^2}\\ &=\frac {5}{8} a^2 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {5 a b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac {b (a+b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{4 c}-\frac {5 a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{8 b^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 103, normalized size = 0.79 \begin {gather*} \frac {\sqrt {c \left (x^2-\frac {a^2}{b^2}\right )} \left (15 a^3 \sin ^{-1}\left (\frac {b x}{a}\right )+\sqrt {1-\frac {b^2 x^2}{a^2}} \left (-16 a^3+9 a^2 b x+16 a b^2 x^2+6 b^3 x^3\right )\right )}{24 b \sqrt {1-\frac {b^2 x^2}{a^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 101, normalized size = 0.78 \begin {gather*} \frac {5 a^4 \sqrt {c} \log \left (\sqrt {c x^2-\frac {a^2 c}{b^2}}-\sqrt {c} x\right )}{8 b^2}+\frac {\left (-16 a^3+9 a^2 b x+16 a b^2 x^2+6 b^3 x^3\right ) \sqrt {c x^2-\frac {a^2 c}{b^2}}}{24 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 238, normalized size = 1.83 \begin {gather*} \left [\frac {15 \, a^{4} \sqrt {c} \log \left (2 \, b^{2} c x^{2} - 2 \, b^{2} \sqrt {c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}} - a^{2} c\right ) + 2 \, {\left (6 \, b^{4} x^{3} + 16 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x - 16 \, a^{3} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{48 \, b^{2}}, \frac {15 \, a^{4} \sqrt {-c} \arctan \left (\frac {b^{2} \sqrt {-c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (6 \, b^{4} x^{3} + 16 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x - 16 \, a^{3} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{24 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 101, normalized size = 0.78 \begin {gather*} \frac {{\left (\frac {15 \, a^{4} \sqrt {c} \log \left ({\left | -\sqrt {b^{2} c} x + \sqrt {b^{2} c x^{2} - a^{2} c} \right |}\right )}{{\left | b \right |}} - \sqrt {b^{2} c x^{2} - a^{2} c} {\left (\frac {16 \, a^{3}}{b} - {\left (9 \, a^{2} + 2 \, {\left (3 \, b^{2} x + 8 \, a b\right )} x\right )} x\right )}\right )} {\left | b \right |}}{24 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 113, normalized size = 0.87 \begin {gather*} -\frac {5 a^{4} \sqrt {c}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}-\frac {a^{2} c}{b^{2}}}\right )}{8 b^{2}}+\frac {5 \sqrt {c \,x^{2}-\frac {a^{2} c}{b^{2}}}\, a^{2} x}{8}+\frac {\left (c \,x^{2}-\frac {a^{2} c}{b^{2}}\right )^{\frac {3}{2}} b^{2} x}{4 c}+\frac {2 \left (\frac {\left (b^{2} x^{2}-a^{2}\right ) c}{b^{2}}\right )^{\frac {3}{2}} a b}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 113, normalized size = 0.87 \begin {gather*} \frac {5}{8} \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} a^{2} x + \frac {{\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} b^{2} x}{4 \, c} - \frac {5 \, a^{4} \sqrt {c} \log \left (2 \, c x + 2 \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} \sqrt {c}\right )}{8 \, b^{2}} + \frac {2 \, {\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} a b}{3 \, c} \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 \sqrt {c\,x^2-\frac {a^2\,c}{b^2}}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.26, size = 408, normalized size = 3.14 \begin {gather*} a^{2} \left (\begin {cases} - \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {b x}{a} \right )}}{2 b^{2}} - \frac {a \sqrt {c} x}{2 b \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} + \frac {b \sqrt {c} x^{3}}{2 a \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} & \text {for}\: \left |{\frac {b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {b x}{a} \right )}}{2 b^{2}} + \frac {i a \sqrt {c} x \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}}{2 b} & \text {otherwise} \end {cases}\right ) + 2 a b \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\left (- \frac {a^{2} c}{b^{2}} + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} - \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {b x}{a} \right )}}{8 b^{4}} + \frac {a^{3} \sqrt {c} x}{8 b^{3} \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} - \frac {3 a \sqrt {c} x^{3}}{8 b \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} + \frac {b \sqrt {c} x^{5}}{4 a \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} & \text {for}\: \left |{\frac {b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac {i a^{4} \sqrt {c} \operatorname {asin}{\left (\frac {b x}{a} \right )}}{8 b^{4}} - \frac {i a^{3} \sqrt {c} x}{8 b^{3} \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} + \frac {3 i a \sqrt {c} x^{3}}{8 b \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} - \frac {i b \sqrt {c} x^{5}}{4 a \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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