Optimal. Leaf size=28 \[ -\frac {2 c \sqrt {\frac {c}{(a+b x)^3}}}{7 b (a+b x)^2} \]
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Rubi [A] time = 0.01, antiderivative size = 28, 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} \[ -\frac {2 c \sqrt {\frac {c}{(a+b x)^3}}}{7 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \left (\frac {c}{(a+b x)^3}\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {c}{x^3}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\left (c \sqrt {\frac {c}{(a+b x)^3}} (a+b x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{9/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {2 c \sqrt {\frac {c}{(a+b x)^3}}}{7 b (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.89 \[ -\frac {2 (a+b x) \left (\frac {c}{(a+b x)^3}\right )^{3/2}}{7 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 58, normalized size = 2.07 \[ -\frac {2 \, c \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{7 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 52, normalized size = 1.86 \[ -\frac {2 \, c^{5} \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )}{7 \, {\left (b c x + a c\right )}^{\frac {7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.79 \[ -\frac {2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {3}{2}}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 24, normalized size = 0.86 \[ -\frac {2 \, {\left (b c^{\frac {3}{2}} x + a c^{\frac {3}{2}}\right )}}{7 \, {\left (b x + a\right )}^{\frac {9}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 24, normalized size = 0.86 \[ -\frac {2\,c\,\sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}}{7\,b\,{\left (a+b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.92, size = 357, normalized size = 12.75 \[ \begin {cases} - \frac {67 a^{3} c^{\frac {3}{2}} \left (\frac {1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} - \frac {201 a^{2} b c^{\frac {3}{2}} x \left (\frac {1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} - \frac {201 a b^{2} c^{\frac {3}{2}} x^{2} \left (\frac {1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} - \frac {67 b^{3} c^{\frac {3}{2}} x^{3} \left (\frac {1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} + \frac {63 c^{\frac {3}{2}} \sqrt {\frac {1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} & \text {for}\: b \neq 0 \\x \left (\frac {c}{a^{3}}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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