Optimal. Leaf size=30 \[ -\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \]
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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} \[ -\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (c x^3\right )^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x)^{3/2} \operatorname {Subst}\left (\int \frac {1}{x^{15/2}} \, dx,x,a+b x\right )}{b c^2 \sqrt {c (a+b x)^3}}\\ &=-\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.83 \[ -\frac {2 (a+b x)}{13 b \left (c (a+b x)^3\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 151, normalized size = 5.03 \[ -\frac {2 \, \sqrt {b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{13 \, {\left (b^{9} c^{3} x^{8} + 8 \, a b^{8} c^{3} x^{7} + 28 \, a^{2} b^{7} c^{3} x^{6} + 56 \, a^{3} b^{6} c^{3} x^{5} + 70 \, a^{4} b^{5} c^{3} x^{4} + 56 \, a^{5} b^{4} c^{3} x^{3} + 28 \, a^{6} b^{3} c^{3} x^{2} + 8 \, a^{7} b^{2} c^{3} x + a^{8} b c^{3}\right )}} \]
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 = 22, normalized size = 0.73 \[ -\frac {2 \left (b x +a \right )}{13 \left (\left (b x +a \right )^{3} c \right )^{\frac {5}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.79, size = 85, normalized size = 2.83 \[ -\frac {2 \, \sqrt {c}}{13 \, {\left (b^{6} c^{3} x^{5} + 5 \, a b^{5} c^{3} x^{4} + 10 \, a^{2} b^{4} c^{3} x^{3} + 10 \, a^{3} b^{3} c^{3} x^{2} + 5 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )} {\left (b x + a\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 26, normalized size = 0.87 \[ -\frac {2\,\sqrt {c\,{\left (a+b\,x\right )}^3}}{13\,b\,c^3\,{\left (a+b\,x\right )}^8} \]
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 \[ \int \frac {1}{\left (c \left (a + b x\right )^{3}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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