Optimal. Leaf size=261 \[ -\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {b \left (2 a-5 b \sqrt {c}\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 \left (a-b \sqrt {c}\right )^{5/2} c^{3/2}}-\frac {b \left (2 a+5 b \sqrt {c}\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 \left (a+b \sqrt {c}\right )^{5/2} c^{3/2}} \]
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Rubi [A]
time = 0.35, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {378, 1412, 837,
841, 1180, 213} \begin {gather*} -\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 x^2 \left (a^2-b^2 c\right )}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c x \left (a^2-b^2 c\right )^2}+\frac {b d^2 \left (2 a-5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 c^{3/2} \left (a-b \sqrt {c}\right )^{5/2}}-\frac {b d^2 \left (2 a+5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 c^{3/2} \left (a+b \sqrt {c}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 378
Rule 837
Rule 841
Rule 1180
Rule 1412
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a+b \sqrt {c+d x}}} \, dx &=d^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}} (-c+x)^3} \, dx,x,c+d x\right )\\ &=\left (2 d^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x} \left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}+\frac {d^2 \text {Subst}\left (\int \frac {-\frac {1}{2} a b c+\frac {5}{2} b^2 c x}{\sqrt {a+b x} \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \text {Subst}\left (\int \frac {\frac {1}{2} a b c \left (a^2-4 b^2 c\right )+\frac {1}{4} b^2 c \left (a^2+5 b^2 c\right ) x}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{4 c^2 \left (a^2-b^2 c\right )^2}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \text {Subst}\left (\int \frac {\frac {1}{2} a b^2 c \left (a^2-4 b^2 c\right )-\frac {1}{4} a b^2 c \left (a^2+5 b^2 c\right )+\frac {1}{4} b^2 c \left (a^2+5 b^2 c\right ) x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 c^2 \left (a^2-b^2 c\right )^2}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}-\frac {\left (b \left (2 a-5 b \sqrt {c}\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{16 \left (a-b \sqrt {c}\right )^2 c^{3/2}}+\frac {\left (b \left (2 a+5 b \sqrt {c}\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{16 \left (a+b \sqrt {c}\right )^2 c^{3/2}}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {b \left (2 a-5 b \sqrt {c}\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 \left (a-b \sqrt {c}\right )^{5/2} c^{3/2}}-\frac {b \left (2 a+5 b \sqrt {c}\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 \left (a+b \sqrt {c}\right )^{5/2} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.34, size = 249, normalized size = 0.95 \begin {gather*} \frac {-\frac {2 \sqrt {c} \sqrt {a+b \sqrt {c+d x}} \left (4 a^3 c+b^3 c (4 c-5 d x) \sqrt {c+d x}-a^2 b \sqrt {c+d x} (4 c+d x)+2 a b^2 c (-2 c+3 d x)\right )}{\left (a^2-b^2 c\right )^2 x^2}+\frac {b \left (2 a+5 b \sqrt {c}\right ) d^2 \tan ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\left (-a-b \sqrt {c}\right )^{5/2}}+\frac {b \left (-2 a+5 b \sqrt {c}\right ) d^2 \tan ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\left (-a+b \sqrt {c}\right )^{5/2}}}{16 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs.
\(2(209)=418\).
time = 0.21, size = 427, normalized size = 1.64
method | result | size |
derivativedivides | \(4 d^{2} b^{4} \left (-\frac {\frac {-\frac {\left (5 \sqrt {b^{2} c}+2 a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{4 \left (b^{2} c +2 a \sqrt {b^{2} c}+a^{2}\right )}+\frac {\left (7 \sqrt {b^{2} c}+2 a \right ) \sqrt {a +b \sqrt {d x +c}}}{4 \sqrt {b^{2} c}+4 a}}{\left (-b \sqrt {d x +c}+\sqrt {b^{2} c}\right )^{2}}-\frac {\left (5 \sqrt {b^{2} c}+2 a \right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{4 \left (b^{2} c +2 a \sqrt {b^{2} c}+a^{2}\right ) \sqrt {-\sqrt {b^{2} c}-a}}}{16 b^{2} c \sqrt {b^{2} c}}+\frac {\frac {-\frac {\left (-5 \sqrt {b^{2} c}+2 a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{4 \left (b^{2} c -2 a \sqrt {b^{2} c}+a^{2}\right )}+\frac {\left (-7 \sqrt {b^{2} c}+2 a \right ) \sqrt {a +b \sqrt {d x +c}}}{-4 \sqrt {b^{2} c}+4 a}}{\left (-b \sqrt {d x +c}-\sqrt {b^{2} c}\right )^{2}}-\frac {\left (5 \sqrt {b^{2} c}-2 a \right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{4 \left (-b^{2} c +2 a \sqrt {b^{2} c}-a^{2}\right ) \sqrt {\sqrt {b^{2} c}-a}}}{16 b^{2} c \sqrt {b^{2} c}}\right )\) | \(427\) |
default | \(4 d^{2} b^{4} \left (-\frac {\frac {-\frac {\left (5 \sqrt {b^{2} c}+2 a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{4 \left (b^{2} c +2 a \sqrt {b^{2} c}+a^{2}\right )}+\frac {\left (7 \sqrt {b^{2} c}+2 a \right ) \sqrt {a +b \sqrt {d x +c}}}{4 \sqrt {b^{2} c}+4 a}}{\left (-b \sqrt {d x +c}+\sqrt {b^{2} c}\right )^{2}}-\frac {\left (5 \sqrt {b^{2} c}+2 a \right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{4 \left (b^{2} c +2 a \sqrt {b^{2} c}+a^{2}\right ) \sqrt {-\sqrt {b^{2} c}-a}}}{16 b^{2} c \sqrt {b^{2} c}}+\frac {\frac {-\frac {\left (-5 \sqrt {b^{2} c}+2 a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{4 \left (b^{2} c -2 a \sqrt {b^{2} c}+a^{2}\right )}+\frac {\left (-7 \sqrt {b^{2} c}+2 a \right ) \sqrt {a +b \sqrt {d x +c}}}{-4 \sqrt {b^{2} c}+4 a}}{\left (-b \sqrt {d x +c}-\sqrt {b^{2} c}\right )^{2}}-\frac {\left (5 \sqrt {b^{2} c}-2 a \right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{4 \left (-b^{2} c +2 a \sqrt {b^{2} c}-a^{2}\right ) \sqrt {\sqrt {b^{2} c}-a}}}{16 b^{2} c \sqrt {b^{2} c}}\right )\) | \(427\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 4390 vs.
\(2 (212) = 424\).
time = 1.28, size = 4390, normalized size = 16.82 \begin {gather*} \text {Too large to display} \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^{3} \sqrt {a + b \sqrt {c + d x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1303 vs.
\(2 (212) = 424\).
time = 3.97, size = 1303, normalized size = 4.99 \begin {gather*} \frac {\frac {{\left ({\left (b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} + a^{4} b c\right )}^{2} {\left (5 \, b^{6} c + a^{2} b^{4}\right )} d^{3} - {\left (13 \, a b^{10} c^{\frac {7}{2}} - 27 \, a^{3} b^{8} c^{\frac {5}{2}} + 15 \, a^{5} b^{6} c^{\frac {3}{2}} - a^{7} b^{4} \sqrt {c}\right )} d^{3} {\left | b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} + a^{4} b c \right |} + 2 \, {\left (4 \, a^{2} b^{14} c^{6} - 17 \, a^{4} b^{12} c^{5} + 28 \, a^{6} b^{10} c^{4} - 22 \, a^{8} b^{8} c^{3} + 8 \, a^{10} b^{6} c^{2} - a^{12} b^{4} c\right )} d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{4} c^{3} - 2 \, a^{3} b^{2} c^{2} + a^{5} c + \sqrt {{\left (a b^{4} c^{3} - 2 \, a^{3} b^{2} c^{2} + a^{5} c\right )}^{2} + {\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )}}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}}}\right )}{{\left (b^{9} c^{6} - a b^{8} c^{\frac {11}{2}} - 4 \, a^{2} b^{7} c^{5} + 4 \, a^{3} b^{6} c^{\frac {9}{2}} + 6 \, a^{4} b^{5} c^{4} - 6 \, a^{5} b^{4} c^{\frac {7}{2}} - 4 \, a^{6} b^{3} c^{3} + 4 \, a^{7} b^{2} c^{\frac {5}{2}} + a^{8} b c^{2} - a^{9} c^{\frac {3}{2}}\right )} \sqrt {-b \sqrt {c} - a} {\left | b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} + a^{4} b c \right |}} + \frac {{\left ({\left (b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} + a^{4} b c\right )}^{2} {\left (5 \, b^{6} c + a^{2} b^{4}\right )} d^{3} + {\left (13 \, a b^{10} c^{\frac {7}{2}} - 27 \, a^{3} b^{8} c^{\frac {5}{2}} + 15 \, a^{5} b^{6} c^{\frac {3}{2}} - a^{7} b^{4} \sqrt {c}\right )} d^{3} {\left | b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} + a^{4} b c \right |} + 2 \, {\left (4 \, a^{2} b^{14} c^{6} - 17 \, a^{4} b^{12} c^{5} + 28 \, a^{6} b^{10} c^{4} - 22 \, a^{8} b^{8} c^{3} + 8 \, a^{10} b^{6} c^{2} - a^{12} b^{4} c\right )} d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{4} c^{3} - 2 \, a^{3} b^{2} c^{2} + a^{5} c - \sqrt {{\left (a b^{4} c^{3} - 2 \, a^{3} b^{2} c^{2} + a^{5} c\right )}^{2} + {\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )}}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}}}\right )}{{\left (b^{9} c^{6} + a b^{8} c^{\frac {11}{2}} - 4 \, a^{2} b^{7} c^{5} - 4 \, a^{3} b^{6} c^{\frac {9}{2}} + 6 \, a^{4} b^{5} c^{4} + 6 \, a^{5} b^{4} c^{\frac {7}{2}} - 4 \, a^{6} b^{3} c^{3} - 4 \, a^{7} b^{2} c^{\frac {5}{2}} + a^{8} b c^{2} + a^{9} c^{\frac {3}{2}}\right )} \sqrt {b \sqrt {c} - a} {\left | b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} + a^{4} b c \right |}} - \frac {2 \, {\left (9 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{8} c^{2} d^{3} - 19 \, \sqrt {\sqrt {d x + c} b + a} a b^{8} c^{2} d^{3} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{6} c d^{3} + 21 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{6} c d^{3} - 30 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{6} c d^{3} + 18 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{6} c d^{3} - {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} b^{4} d^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} b^{4} d^{3} - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} b^{4} d^{3} + \sqrt {\sqrt {d x + c} b + a} a^{5} b^{4} d^{3}\right )}}{{\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\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 )}^{2}}}{16 \, b^{2} d} \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 {1}{x^3\,\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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