Optimal. Leaf size=66 \[ -\frac {a^2}{5 c^2 x^4 \sqrt {c x^2}}-\frac {a b}{2 c^2 x^3 \sqrt {c x^2}}-\frac {b^2}{3 c^2 x^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, 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}{5 c^2 x^4 \sqrt {c x^2}}-\frac {a b}{2 c^2 x^3 \sqrt {c x^2}}-\frac {b^2}{3 c^2 x^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^6} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^6}+\frac {2 a b}{x^5}+\frac {b^2}{x^4}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a^2}{5 c^2 x^4 \sqrt {c x^2}}-\frac {a b}{2 c^2 x^3 \sqrt {c x^2}}-\frac {b^2}{3 c^2 x^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 0.58 \begin {gather*} -\frac {\sqrt {c x^2} \left (6 a^2+15 a b x+10 b^2 x^2\right )}{30 c^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.93, size = 27, normalized size = 0.41 \begin {gather*} \frac {-\frac {a^2}{5}-\frac {a b x}{2}-\frac {b^2 x^2}{3}}{{\left (c x^2\right )}^{\frac {5}{2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 29, normalized size = 0.44
method | result | size |
gosper | \(-\frac {10 x^{2} b^{2}+15 a b x +6 a^{2}}{30 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(29\) |
default | \(-\frac {10 x^{2} b^{2}+15 a b x +6 a^{2}}{30 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(29\) |
risch | \(\frac {-\frac {1}{3} x^{2} b^{2}-\frac {1}{2} a b x -\frac {1}{5} a^{2}}{c^{2} x^{4} \sqrt {c \,x^{2}}}\) | \(34\) |
trager | \(\frac {\left (-1+x \right ) \left (6 a^{2} x^{4}+15 a b \,x^{4}+10 b^{2} x^{4}+6 a^{2} x^{3}+15 a b \,x^{3}+10 b^{2} x^{3}+6 a^{2} x^{2}+15 a b \,x^{2}+10 x^{2} b^{2}+6 a^{2} x +15 a b x +6 a^{2}\right ) \sqrt {c \,x^{2}}}{30 c^{3} x^{6}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 37, normalized size = 0.56 \begin {gather*} -\frac {b^{2}}{3 \, \left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a b}{2 \, c^{\frac {5}{2}} x^{4}} - \frac {a^{2}}{5 \, c^{\frac {5}{2}} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 34, normalized size = 0.52 \begin {gather*} -\frac {{\left (10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}\right )} \sqrt {c x^{2}}}{30 \, c^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 46, normalized size = 0.70 \begin {gather*} - \frac {a^{2}}{5 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {a b x}{2 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {b^{2} x^{2}}{3 \left (c x^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 41, normalized size = 0.62 \begin {gather*} \frac {-10 b^{2} x^{2}-15 a b x-6 a^{2}}{\sqrt {c}\cdot 30 \left (c^{2} x^{5} \mathrm {sign}\left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 42, normalized size = 0.64 \begin {gather*} -\frac {6\,a^2\,\sqrt {x^2}+10\,b^2\,x^2\,\sqrt {x^2}+15\,a\,b\,x\,\sqrt {x^2}}{30\,c^{5/2}\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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