Optimal. Leaf size=70 \[ -\frac {a x^2}{b^2 c \sqrt {c x^2}}+\frac {x^3}{2 b c \sqrt {c x^2}}+\frac {a^2 x \log (a+b x)}{b^3 c \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45}
\begin {gather*} \frac {a^2 x \log (a+b x)}{b^3 c \sqrt {c x^2}}-\frac {a x^2}{b^2 c \sqrt {c x^2}}+\frac {x^3}{2 b c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {x^5}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {x^2}{a+b x} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a x^2}{b^2 c \sqrt {c x^2}}+\frac {x^3}{2 b c \sqrt {c x^2}}+\frac {a^2 x \log (a+b x)}{b^3 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.59 \begin {gather*} \frac {x^3 \left (b x (-2 a+b x)+2 a^2 \log (a+b x)\right )}{2 b^3 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 40, normalized size = 0.57
method | result | size |
default | \(\frac {x^{3} \left (x^{2} b^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{3}}\) | \(40\) |
risch | \(\frac {x \left (\frac {1}{2} x^{2} b -a x \right )}{c \sqrt {c \,x^{2}}\, b^{2}}+\frac {a^{2} x \ln \left (b x +a \right )}{b^{3} c \sqrt {c \,x^{2}}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs.
\(2 (62) = 124\).
time = 0.31, size = 140, normalized size = 2.00 \begin {gather*} \frac {x^{3}}{2 \, \sqrt {c x^{2}} b c} - \frac {a x^{2}}{\sqrt {c x^{2}} b^{2} c} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3} c^{\frac {3}{2}}} - \frac {7 \, a^{2} x}{2 \, \sqrt {c x^{2}} b^{3} c} + \frac {a^{2} \log \left (b x\right )}{b^{3} c^{\frac {3}{2}}} + \frac {2 \, a^{3}}{\sqrt {c x^{2}} b^{4} c} - \frac {2 \, a^{3}}{b^{4} c^{\frac {3}{2}} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 42, normalized size = 0.60 \begin {gather*} \frac {{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} c^{2} x} \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 \frac {x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.48, size = 72, normalized size = 1.03 \begin {gather*} -\frac {\frac {2 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3} \sqrt {c}} - \frac {2 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {b \sqrt {c} x^{2} \mathrm {sgn}\left (x\right ) - 2 \, a \sqrt {c} x \mathrm {sgn}\left (x\right )}{b^{2} c}}{2 \, 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 \frac {x^5}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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