Optimal. Leaf size=30 \[ \frac {2 (a+b x)^7}{17 b c^2 \sqrt {\frac {c}{(a+b x)^3}}} \]
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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} \[ \frac {2 (a+b x)^7}{17 b c^2 \sqrt {\frac {c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (\frac {c}{x^3}\right )^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x^{15/2} \, dx,x,a+b x\right )}{b c^2 \sqrt {\frac {c}{(a+b x)^3}} (a+b x)^{3/2}}\\ &=\frac {2 (a+b x)^7}{17 b c^2 \sqrt {\frac {c}{(a+b x)^3}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.83 \[ \frac {2 (a+b x)}{17 b \left (\frac {c}{(a+b x)^3}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 145, normalized size = 4.83 \[ \frac {2 \, {\left (b^{10} x^{10} + 10 \, a b^{9} x^{9} + 45 \, a^{2} b^{8} x^{8} + 120 \, a^{3} b^{7} x^{7} + 210 \, a^{4} b^{6} x^{6} + 252 \, a^{5} b^{5} x^{5} + 210 \, a^{6} b^{4} x^{4} + 120 \, a^{7} b^{3} x^{3} + 45 \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10}\right )} \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{17 \, b c^{3}} \]
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 \[ \int \frac {1}{\left (\frac {c}{{\left (b x + a\right )}^{3}}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \frac {\frac {2 b x}{17}+\frac {2 a}{17}}{\left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 27, normalized size = 0.90 \[ \frac {2 \, {\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{\frac {15}{2}}}{17 \, b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 152, normalized size = 5.07 \[ \sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}\,\left (\frac {20\,a^9\,x}{17\,c^3}+\frac {2\,a^{10}}{17\,b\,c^3}+\frac {2\,b^9\,x^{10}}{17\,c^3}+\frac {90\,a^8\,b\,x^2}{17\,c^3}+\frac {20\,a\,b^8\,x^9}{17\,c^3}+\frac {240\,a^7\,b^2\,x^3}{17\,c^3}+\frac {420\,a^6\,b^3\,x^4}{17\,c^3}+\frac {504\,a^5\,b^4\,x^5}{17\,c^3}+\frac {420\,a^4\,b^5\,x^6}{17\,c^3}+\frac {240\,a^3\,b^6\,x^7}{17\,c^3}+\frac {90\,a^2\,b^7\,x^8}{17\,c^3}\right ) \]
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 \[ \int \frac {1}{\left (\frac {c}{\left (a + b x\right )^{3}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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