Optimal. Leaf size=30 \[ \frac {(a+b x)^3}{4 b c \sqrt {\frac {c}{(a+b x)^2}}} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac {(a+b x)^3}{4 b c \sqrt {\frac {c}{(a+b x)^2}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (\frac {c}{x^2}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,a+b x\right )}{b c \sqrt {\frac {c}{(a+b x)^2}} (a+b x)}\\ &=\frac {(a+b x)^3}{4 b c \sqrt {\frac {c}{(a+b x)^2}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.83 \[ \frac {a+b x}{4 b \left (\frac {c}{(a+b x)^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 70, normalized size = 2.33 \[ \frac {{\left (b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 4 \, a^{4} x\right )} \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, c^{2}} \]
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 \[ \int \frac {1}{\left (\frac {c}{{\left (b x + a\right )}^{2}}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 51, normalized size = 1.70 \[ \frac {\left (b^{3} x^{3}+4 a \,b^{2} x^{2}+6 a^{2} b x +4 a^{3}\right ) x}{4 \left (b x +a \right )^{3} \left (\frac {c}{\left (b x +a \right )^{2}}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 37, normalized size = 1.23 \[ \frac {b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 70, normalized size = 2.33 \[ \sqrt {\frac {c}{{\left (a+b\,x\right )}^2}}\,\left (\frac {a^4\,x}{c^2}+\frac {b^4\,x^5}{4\,c^2}+\frac {5\,a^3\,b\,x^2}{2\,c^2}+\frac {5\,a\,b^3\,x^4}{4\,c^2}+\frac {5\,a^2\,b^2\,x^3}{2\,c^2}\right ) \]
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 \[ \int \frac {1}{\left (\frac {c}{\left (a + b x\right )^{2}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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