Optimal. Leaf size=66 \[ -\frac {a^2}{6 c x^5 \sqrt {c x^2}}-\frac {2 a b}{5 c x^4 \sqrt {c x^2}}-\frac {b^2}{4 c x^3 \sqrt {c x^2}} \]
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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, 43} \[ -\frac {a^2}{6 c x^5 \sqrt {c x^2}}-\frac {2 a b}{5 c x^4 \sqrt {c x^2}}-\frac {b^2}{4 c x^3 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^4 \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^7} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^7}+\frac {2 a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a^2}{6 c x^5 \sqrt {c x^2}}-\frac {2 a b}{5 c x^4 \sqrt {c x^2}}-\frac {b^2}{4 c x^3 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.53 \[ \frac {-10 a^2-24 a b x-15 b^2 x^2}{60 x^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 0.52 \[ -\frac {{\left (15 \, b^{2} x^{2} + 24 \, a b x + 10 \, a^{2}\right )} \sqrt {c x^{2}}}{60 \, c^{2} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.48 \[ -\frac {15 b^{2} x^{2}+24 a b x +10 a^{2}}{60 \left (c \,x^{2}\right )^{\frac {3}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 33, normalized size = 0.50 \[ -\frac {b^{2}}{4 \, c^{\frac {3}{2}} x^{4}} - \frac {2 \, a b}{5 \, c^{\frac {3}{2}} x^{5}} - \frac {a^{2}}{6 \, c^{\frac {3}{2}} x^{6}} \]
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 \[ -\frac {10\,a^2\,\sqrt {x^2}+15\,b^2\,x^2\,\sqrt {x^2}+24\,a\,b\,x\,\sqrt {x^2}}{60\,c^{3/2}\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 61, normalized size = 0.92 \[ - \frac {a^{2}}{6 c^{\frac {3}{2}} x^{3} \left (x^{2}\right )^{\frac {3}{2}}} - \frac {2 a b}{5 c^{\frac {3}{2}} x^{2} \left (x^{2}\right )^{\frac {3}{2}}} - \frac {b^{2}}{4 c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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