Optimal. Leaf size=102 \[ \frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}+\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {38, 63, 217, 203} \[ \frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}+\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 38
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx &=\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{4} \left (3 a^2 c\right ) \int \sqrt {a+b x} \sqrt {a c-b c x} \, dx\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{8} \left (3 a^4 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {\left (3 a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{4 b}\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {\left (3 a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 109, normalized size = 1.07 \[ \frac {c^2 \left (-6 a^{9/2} \sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+5 a^4 b x-7 a^2 b^3 x^3+2 b^5 x^5\right )}{8 b \sqrt {a+b x} \sqrt {c (a-b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 193, normalized size = 1.89 \[ \left [\frac {3 \, a^{4} \sqrt {-c} c \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) - 2 \, {\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{16 \, b}, -\frac {3 \, a^{4} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{8 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 185, normalized size = 1.81 \[ \frac {3 \sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, a^{4} c^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{8 \sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}+\frac {3 \sqrt {-b c x +a c}\, \sqrt {b x +a}\, a^{3} c}{8 b}+\frac {\left (-b c x +a c \right )^{\frac {3}{2}} \sqrt {b x +a}\, a^{2}}{8 b}-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {5}{2}} a}{4 b c}-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}{4 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 63, normalized size = 0.62 \[ \frac {3 \, a^{4} c^{\frac {3}{2}} \arcsin \left (\frac {b x}{a}\right )}{8 \, b} + \frac {3}{8} \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c x + \frac {1}{4} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} x \]
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 \[ \int {\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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