Optimal. Leaf size=163 \[ -\frac {\sqrt {a+b \sqrt {c+d x}} \left (a-b \sqrt {c+d x}\right )}{x \left (a^2-b^2 c\right )}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} \left (a-b \sqrt {c}\right )^{3/2}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {c} \left (a+b \sqrt {c}\right )^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {371, 1398, 823, 827, 1166, 207} \begin {gather*} -\frac {\sqrt {a+b \sqrt {c+d x}} \left (a-b \sqrt {c+d x}\right )}{x \left (a^2-b^2 c\right )}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} \left (a-b \sqrt {c}\right )^{3/2}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {c} \left (a+b \sqrt {c}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 371
Rule 823
Rule 827
Rule 1166
Rule 1398
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx &=d \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}} (-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x} \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}+\frac {d \operatorname {Subst}\left (\int \frac {-\frac {1}{2} a b c+\frac {1}{2} b^2 c x}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{c \left (a^2-b^2 c\right )}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {-a b^2 c+\frac {1}{2} b^2 c x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{c \left (a^2-b^2 c\right )}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 \left (a-b \sqrt {c}\right ) \sqrt {c}}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 \left (a+b \sqrt {c}\right ) \sqrt {c}}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \left (a-b \sqrt {c}\right )^{3/2} \sqrt {c}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \left (a+b \sqrt {c}\right )^{3/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 216, normalized size = 1.33 \begin {gather*} \frac {\sqrt {a-b \sqrt {c}} \left (b d x \left (a-b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )-2 \sqrt {c} \sqrt {a+b \sqrt {c}} \left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}\right )-b d x \left (a+b \sqrt {c}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} x \sqrt {a-b \sqrt {c}} \sqrt {a+b \sqrt {c}} \left (a^2-b^2 c\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 200, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {a+b \sqrt {c+d x}} \left (a d-b d \sqrt {c+d x}\right )}{d x \left (a^2-b^2 c\right )}+\frac {b d \tan ^{-1}\left (\frac {\sqrt {b \sqrt {c}-a} \sqrt {a+b \sqrt {c+d x}}}{a-b \sqrt {c}}\right )}{2 \sqrt {c} \left (b \sqrt {c}-a\right )^{3/2}}-\frac {b d \tan ^{-1}\left (\frac {\sqrt {-a-b \sqrt {c}} \sqrt {a+b \sqrt {c+d x}}}{a+b \sqrt {c}}\right )}{2 \sqrt {c} \left (-a-b \sqrt {c}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 2493, normalized size = 15.29
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.76, size = 654, normalized size = 4.01 \begin {gather*} \frac {\frac {{\left ({\left (b^{3} c - a^{2} b\right )}^{2} b^{4} c^{\frac {3}{2}} d^{2} - 2 \, {\left (a b^{6} c^{2} - a^{3} b^{4} c\right )} d^{2} {\left | -b^{3} c + a^{2} b \right |} + {\left (a^{2} b^{8} c^{\frac {5}{2}} - 2 \, a^{4} b^{6} c^{\frac {3}{2}} + a^{6} b^{4} \sqrt {c}\right )} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c - a^{3} + \sqrt {{\left (a b^{2} c - a^{3}\right )}^{2} + {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} {\left (b^{2} c - a^{2}\right )}}}{b^{2} c - a^{2}}}}\right )}{{\left (b^{5} c^{\frac {7}{2}} + a b^{4} c^{3} - 2 \, a^{2} b^{3} c^{\frac {5}{2}} - 2 \, a^{3} b^{2} c^{2} + a^{4} b c^{\frac {3}{2}} + a^{5} c\right )} \sqrt {b \sqrt {c} - a} {\left | -b^{3} c + a^{2} b \right |}} + \frac {{\left ({\left (b^{3} c - a^{2} b\right )}^{2} b^{4} c^{\frac {3}{2}} d^{2} + 2 \, {\left (a b^{6} c^{2} - a^{3} b^{4} c\right )} d^{2} {\left | -b^{3} c + a^{2} b \right |} + {\left (a^{2} b^{8} c^{\frac {5}{2}} - 2 \, a^{4} b^{6} c^{\frac {3}{2}} + a^{6} b^{4} \sqrt {c}\right )} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c - a^{3} - \sqrt {{\left (a b^{2} c - a^{3}\right )}^{2} + {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} {\left (b^{2} c - a^{2}\right )}}}{b^{2} c - a^{2}}}}\right )}{{\left (b^{5} c^{\frac {7}{2}} - a b^{4} c^{3} - 2 \, a^{2} b^{3} c^{\frac {5}{2}} + 2 \, a^{3} b^{2} c^{2} + a^{4} b c^{\frac {3}{2}} - a^{5} c\right )} \sqrt {-b \sqrt {c} - a} {\left | -b^{3} c + a^{2} b \right |}} + \frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} d^{2} - 2 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} d^{2}\right )}}{{\left (b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{2} + 2 \, {\left (\sqrt {d x + c} b + a\right )} a - a^{2}\right )} {\left (b^{2} c - a^{2}\right )}}}{2 \, b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 265, normalized size = 1.63 \begin {gather*} \frac {2 \sqrt {b^{2} c}\, d \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{\left (-4 a -4 \sqrt {b^{2} c}\right ) \sqrt {-a -\sqrt {b^{2} c}}\, c}-\frac {2 \sqrt {b^{2} c}\, d \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{\left (-4 a +4 \sqrt {b^{2} c}\right ) \sqrt {-a +\sqrt {b^{2} c}}\, c}-\frac {2 \sqrt {b^{2} c}\, \sqrt {a +\sqrt {d x +c}\, b}\, d}{\left (-4 a -4 \sqrt {b^{2} c}\right ) \left (-\sqrt {d x +c}\, b +\sqrt {b^{2} c}\right ) c}-\frac {2 \sqrt {b^{2} c}\, \sqrt {a +\sqrt {d x +c}\, b}\, d}{\left (-4 a +4 \sqrt {b^{2} c}\right ) \left (\sqrt {d x +c}\, b +\sqrt {b^{2} c}\right ) c} \end {gather*}
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*} \int \frac {1}{\sqrt {\sqrt {d x + c} b + a} x^{2}}\,{d x} \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 {1}{x^2\,\sqrt {a+b\,\sqrt {c+d\,x}}} \,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 {1}{x^{2} \sqrt {a + b \sqrt {c + d x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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