Optimal. Leaf size=97 \[ \frac {b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right )}{6 a^{3/2}}-\frac {b c \sqrt {a+b \sqrt {c x^3}}}{6 a \sqrt {c x^3}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{3 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {369, 266, 47, 51, 63, 208} \begin {gather*} \frac {b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right )}{6 a^{3/2}}-\frac {b c \sqrt {a+b \sqrt {c x^3}}}{6 a \sqrt {c x^3}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^4} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x^4} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x}}{x^3} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{3 x^3}+\operatorname {Subst}\left (\frac {1}{6} \left (b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b \sqrt {c} x}} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{3 x^3}-\frac {b c \sqrt {a+b \sqrt {c x^3}}}{6 a \sqrt {c x^3}}-\operatorname {Subst}\left (\frac {\left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,x^{3/2}\right )}{12 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{3 x^3}-\frac {b c \sqrt {a+b \sqrt {c x^3}}}{6 a \sqrt {c x^3}}-\operatorname {Subst}\left (\frac {\left (b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+b \sqrt {c} x^{3/2}}\right )}{6 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{3 x^3}-\frac {b c \sqrt {a+b \sqrt {c x^3}}}{6 a \sqrt {c x^3}}+\frac {b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right )}{6 a^{3/2}}\\ \end {align*}
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Mathematica [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.27, size = 80, normalized size = 0.82 \begin {gather*} \frac {b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right )}{6 a^{3/2}}-\frac {\sqrt {a+b \sqrt {c x^3}} \left (2 a+b \sqrt {c x^3}\right )}{6 a x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 122, normalized size = 1.26 \begin {gather*} -\frac {{\left (\frac {b^{3} c^{3} \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{2} x + a c^{2}}}{\sqrt {-a} c}\right )}{\sqrt {-a} a} + \frac {\sqrt {\sqrt {c x} b c^{2} x + a c^{2}} a b^{3} c^{6} + {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {3}{2}} b^{3} c^{4}}{a b^{2} c^{5} x^{3}}\right )} {\left | c \right |}}{6 \, b c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 81, normalized size = 0.84 \begin {gather*} -\frac {-a \,b^{2} c \,x^{3} \arctanh \left (\frac {\sqrt {a +\sqrt {c \,x^{3}}\, b}}{\sqrt {a}}\right )+2 \sqrt {a +\sqrt {c \,x^{3}}\, b}\, a^{\frac {5}{2}}+\sqrt {c \,x^{3}}\, \sqrt {a +\sqrt {c \,x^{3}}\, b}\, a^{\frac {3}{2}} b}{6 a^{\frac {5}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 126, normalized size = 1.30 \begin {gather*} -\frac {1}{12} \, {\left (\frac {b^{2} \log \left (\frac {\sqrt {\sqrt {c x^{3}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{3}} b + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {c x^{3}} b + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {\sqrt {c x^{3}} b + a} a b^{2}\right )}}{{\left (\sqrt {c x^{3}} b + a\right )}^{2} a - 2 \, {\left (\sqrt {c x^{3}} b + a\right )} a^{2} + a^{3}}\right )} c \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x^4} \,d x \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 {a + b \sqrt {c x^{3}}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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