Optimal. Leaf size=417 \[ -\frac {2 \sqrt [3]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {4 d \sqrt [3]{c+d x}}{9 b (b c-a d) \sqrt {a+b x}}+\frac {4 \sqrt {2-\sqrt {3}} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} b^{4/3} (b c-a d) \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Rubi [A]
time = 0.26, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {49, 53, 65, 225}
\begin {gather*} \frac {4 \sqrt {2-\sqrt {3}} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} b^{4/3} \sqrt {a+b x} (b c-a d) \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {4 d \sqrt [3]{c+d x}}{9 b \sqrt {a+b x} (b c-a d)}-\frac {2 \sqrt [3]{c+d x}}{3 b (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 53
Rule 65
Rule 225
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{5/2}} \, dx &=-\frac {2 \sqrt [3]{c+d x}}{3 b (a+b x)^{3/2}}+\frac {(2 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{2/3}} \, dx}{9 b}\\ &=-\frac {2 \sqrt [3]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {4 d \sqrt [3]{c+d x}}{9 b (b c-a d) \sqrt {a+b x}}-\frac {\left (2 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx}{27 b (b c-a d)}\\ &=-\frac {2 \sqrt [3]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {4 d \sqrt [3]{c+d x}}{9 b (b c-a d) \sqrt {a+b x}}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{9 b (b c-a d)}\\ &=-\frac {2 \sqrt [3]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {4 d \sqrt [3]{c+d x}}{9 b (b c-a d) \sqrt {a+b x}}+\frac {4 \sqrt {2-\sqrt {3}} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} b^{4/3} (b c-a d) \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 73, normalized size = 0.18 \begin {gather*} -\frac {2 \sqrt [3]{c+d x} \, _2F_1\left (-\frac {3}{2},-\frac {1}{3};-\frac {1}{2};\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt [3]{c + d x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{1/3}}{{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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