Optimal. Leaf size=172 \[ \frac {4 a^5 \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^6 c^3}-\frac {20 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac {8 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 172, 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} \[ -\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}+\frac {8 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}-\frac {20 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac {4 a^5 \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x^4} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\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 \frac {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}{b^5 c^{5/2} \sqrt {a+b \sqrt {c} x}}+\frac {5 a^4 \sqrt {a+b \sqrt {c} x}}{b^5 c^{5/2}}-\frac {10 a^3 \left (a+b \sqrt {c} x\right )^{3/2}}{b^5 c^{5/2}}+\frac {10 a^2 \left (a+b \sqrt {c} x\right )^{5/2}}{b^5 c^{5/2}}-\frac {5 a \left (a+b \sqrt {c} x\right )^{7/2}}{b^5 c^{5/2}}+\frac {\left (a+b \sqrt {c} x\right )^{9/2}}{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 \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^6 c^3}-\frac {20 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac {8 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}+\frac {20 a \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 )^{11/2}}{11 b^6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 111, normalized size = 0.65 \[ \frac {4 \sqrt {a+b \sqrt {\frac {c}{x}}} \left (256 a^5 x^2-128 a^4 b x^2 \sqrt {\frac {c}{x}}+96 a^3 b^2 c x-80 a^2 b^3 c x \sqrt {\frac {c}{x}}+70 a b^4 c^2-63 b^5 c x \left (\frac {c}{x}\right )^{3/2}\right )}{693 b^6 c^3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 89, normalized size = 0.52 \[ \frac {4 \, {\left (70 \, a b^{4} c^{2} + 96 \, a^{3} b^{2} c x + 256 \, a^{5} x^{2} - {\left (63 \, b^{5} c^{2} + 80 \, a^{2} b^{3} c x + 128 \, a^{4} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{693 \, b^{6} c^{3} x^{2}} \]
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 \[ \int \frac {1}{\sqrt {b \sqrt {\frac {c}{x}} + a} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 400, normalized size = 2.33 \[ -\frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (-693 \sqrt {\frac {c}{x}}\, a^{6} b \,x^{4} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+693 \sqrt {\frac {c}{x}}\, a^{6} b \,x^{4} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+1386 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {13}{2}} x^{\frac {7}{2}}+1386 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, a^{\frac {13}{2}} x^{\frac {7}{2}}-2772 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {5}{2}}+1748 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \sqrt {\frac {c}{x}}\, a^{\frac {9}{2}} b \,x^{\frac {5}{2}}-1236 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} c \,x^{\frac {3}{2}}+852 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \left (\frac {c}{x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}}-532 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{4} c^{2} \sqrt {x}+252 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \left (\frac {c}{x}\right )^{\frac {5}{2}} \sqrt {a}\, b^{5} x^{\frac {5}{2}}\right )}{693 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \left (\frac {c}{x}\right )^{\frac {7}{2}} \sqrt {a}\, b^{7} x^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 127, normalized size = 0.74 \[ -\frac {4 \, {\left (\frac {63 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {11}{2}}}{b^{6}} - \frac {385 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {9}{2}} a}{b^{6}} + \frac {990 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} a^{2}}{b^{6}} - \frac {1386 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a^{3}}{b^{6}} + \frac {1155 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{4}}{b^{6}} - \frac {693 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{5}}{b^{6}}\right )}}{693 \, c^{3}} \]
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 \[ \int \frac {1}{x^4\,\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \,d x \]
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 \[ \int \frac {1}{x^{4} \sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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