Optimal. Leaf size=102 \[ \frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b} \]
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Rubi [A]
time = 0.02, 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, 65, 223,
209} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 65
Rule 209
Rule 223
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 ) \text {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 ) \text {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 = 0.12, size = 92, normalized size = 0.90 \begin {gather*} \frac {(c (a-b x))^{3/2} \left (b x \sqrt {a-b x} \sqrt {a+b x} \left (5 a^2-2 b^2 x^2\right )+6 a^4 \tan ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{8 b (a-b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs.
\(2(80)=160\).
time = 0.17, size = 184, normalized size = 1.80
method | result | size |
risch | \(\frac {x \left (-2 x^{2} b^{2}+5 a^{2}\right ) \sqrt {b x +a}\, \left (-b x +a \right ) c^{2}}{8 \sqrt {-c \left (b x -a \right )}}+\frac {3 a^{4} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{8 \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(125\) |
default | \(-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}{4 b c}+\frac {3 a \left (-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {5}{2}}}{3 b c}+\frac {a \left (\frac {\left (-b c x +a c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}+\frac {3 a c \left (\frac {\sqrt {-b c x +a c}\, \sqrt {b x +a}}{b}+\frac {a c \sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}\right )}{2}\right )}{3}\right )}{4}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 63, normalized size = 0.62 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 193, normalized size = 1.89 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} \int \left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs.
\(2 (82) = 164\).
time = 0.06, size = 516, normalized size = 5.06 \begin {gather*} \frac {2 a^{2} c \left (2 \left (\frac {1}{8} \sqrt {a+b x} \sqrt {a+b x}-\frac {12}{32} a\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}+\frac {2 a^{2} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{4 \sqrt {-c}}\right )-2 c \left (2 \left (\left (\left (\frac {1}{16} \sqrt {a+b x} \sqrt {a+b x}-\frac {24960}{92160} a\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {41280}{92160} a^{2}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {37440}{92160} a^{3}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}+\frac {6 a^{4} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{16 \sqrt {-c}}\right )+2 a^{3} c \left (\frac {1}{2} \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}-\frac {2 a c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{2 \sqrt {-c}}\right )-2 a c \left (2 \left (\left (\frac {1}{12} \sqrt {a+b x} \sqrt {a+b x}-\frac {84}{288} a\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {108}{288} a^{2}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}-\frac {2 a^{3} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{4 \sqrt {-c}}\right )}{b} \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 {\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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