Optimal. Leaf size=174 \[ \frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3} \]
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Rubi [A] time = 0.11, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {369, 266, 43} \begin {gather*} -\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{x^4} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int x^5 \sqrt {a+b \sqrt {c} x} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \left (-\frac {a^5 \sqrt {a+b \sqrt {c} x}}{b^5 c^{5/2}}+\frac {5 a^4 \left (a+b \sqrt {c} x\right )^{3/2}}{b^5 c^{5/2}}-\frac {10 a^3 \left (a+b \sqrt {c} x\right )^{5/2}}{b^5 c^{5/2}}+\frac {10 a^2 \left (a+b \sqrt {c} x\right )^{7/2}}{b^5 c^{5/2}}-\frac {5 a \left (a+b \sqrt {c} x\right )^{9/2}}{b^5 c^{5/2}}+\frac {\left (a+b \sqrt {c} x\right )^{11/2}}{b^5 c^{5/2}}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 111, normalized size = 0.64 \begin {gather*} \frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2} \left (256 a^5 x^2-384 a^4 b x^2 \sqrt {\frac {c}{x}}+480 a^3 b^2 c x-560 a^2 b^3 c x \sqrt {\frac {c}{x}}+630 a b^4 c^2-693 b^5 c x \left (\frac {c}{x}\right )^{3/2}\right )}{9009 b^6 c^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 103, normalized size = 0.59 \begin {gather*} \frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2} \left (256 a^5-384 a^4 b \sqrt {\frac {c}{x}}+\frac {480 a^3 b^2 c}{x}-560 a^2 b^3 \left (\frac {c}{x}\right )^{3/2}+\frac {630 a b^4 c^2}{x^2}-693 b^5 \left (\frac {c}{x}\right )^{5/2}\right )}{9009 b^6 c^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 105, normalized size = 0.60 \begin {gather*} -\frac {4 \, {\left (693 \, b^{6} c^{3} - 70 \, a^{2} b^{4} c^{2} x - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} x^{3} + {\left (63 \, a b^{5} c^{2} x + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b x^{3}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{9009 \, b^{6} c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 133, normalized size = 0.76 \begin {gather*} -\frac {4 \sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \left (-256 a^{5} x^{2}+384 \sqrt {\frac {c}{x}}\, a^{4} b \,x^{2}-480 a^{3} b^{2} c x +560 \left (\frac {c}{x}\right )^{\frac {3}{2}} a^{2} b^{3} x^{2}-630 a \,b^{4} c^{2}+693 \left (\frac {c}{x}\right )^{\frac {5}{2}} b^{5} x^{2}\right )}{9009 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, b^{6} c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 127, normalized size = 0.73 \begin {gather*} -\frac {4 \, {\left (\frac {693 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {13}{2}}}{b^{6}} - \frac {4095 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {11}{2}} a}{b^{6}} + \frac {10010 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {9}{2}} a^{2}}{b^{6}} - \frac {12870 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} a^{3}}{b^{6}} + \frac {9009 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {3003 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \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 {\frac {c}{x}}}}{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 {\frac {c}{x}}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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