Optimal. Leaf size=224 \[ \frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c x \left (a^2-b^2 c\right )}-\frac {b d^2 \left (2 a-3 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 c^{3/2} \left (a-b \sqrt {c}\right )^{3/2}}+\frac {b d^2 \left (2 a+3 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 c^{3/2} \left (a+b \sqrt {c}\right )^{3/2}}-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2} \]
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Rubi [A] time = 0.43, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {371, 1398, 821, 12, 741, 827, 1166, 207} \begin {gather*} \frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c x \left (a^2-b^2 c\right )}-\frac {b d^2 \left (2 a-3 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 )^{3/2}}+\frac {b d^2 \left (2 a+3 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 )^{3/2}}-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 371
Rule 741
Rule 821
Rule 827
Rule 1166
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx &=d^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}}}{(-c+x)^3} \, dx,x,c+d x\right )\\ &=\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b x}}{\left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}-\frac {d^2 \operatorname {Subst}\left (\int -\frac {b c}{2 \sqrt {a+b x} \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}+\frac {1}{4} \left (b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}+\frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right ) x}+\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (-2 a^2+3 b^2 c\right )-\frac {a b x}{2}}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}+\frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right ) x}+\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {a^2 b}{2}+\frac {1}{2} b \left (-2 a^2+3 b^2 c\right )-\frac {1}{2} a b x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{4 c \left (a^2-b^2 c\right )}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}+\frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right ) x}+\frac {\left (b \left (2 a-3 b \sqrt {c}\right ) d^2\right ) \operatorname {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 ) c^{3/2}}-\frac {\left (b \left (2 a+3 b \sqrt {c}\right ) d^2\right ) \operatorname {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 ) c^{3/2}}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}+\frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right ) x}-\frac {b \left (2 a-3 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 )^{3/2} c^{3/2}}+\frac {b \left (2 a+3 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 )^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 258, normalized size = 1.15 \begin {gather*} \frac {-\left (a-b \sqrt {c}\right ) \left (2 \sqrt {c} \left (a+b \sqrt {c}\right ) \sqrt {a+b \sqrt {c+d x}} \left (4 a^2 c+a b d x \sqrt {c+d x}-b^2 c (4 c+d x)\right )-b d^2 x^2 \sqrt {a+b \sqrt {c}} \left (2 a^2+a b \sqrt {c}-3 b^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )\right )-b d^2 x^2 \left (2 a-3 b \sqrt {c}\right ) \sqrt {a-b \sqrt {c}} \left (a+b \sqrt {c}\right )^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 c^{3/2} x^2 \left (a^2-b^2 c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.08, size = 310, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {a+b \sqrt {c+d x}} \left (-4 a^2 c d^2-a b d^2 (c+d x)^{3/2}+a b c d^2 \sqrt {c+d x}+3 b^2 c^2 d^2+b^2 c d^2 (c+d x)\right )}{8 c d^2 x^2 \left (b^2 c-a^2\right )}-\frac {d^2 \left (2 a b-3 b^2 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {b \sqrt {c}-a} \sqrt {a+b \sqrt {c+d x}}}{a-b \sqrt {c}}\right )}{16 c^{3/2} \left (a-b \sqrt {c}\right ) \sqrt {b \sqrt {c}-a}}+\frac {d^2 \left (2 a b+3 b^2 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {-a-b \sqrt {c}} \sqrt {a+b \sqrt {c+d x}}}{a+b \sqrt {c}}\right )}{16 c^{3/2} \sqrt {-a-b \sqrt {c}} \left (a+b \sqrt {c}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.31, size = 2856, normalized size = 12.75
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 895, normalized size = 4.00 \begin {gather*} \frac {\frac {{\left ({\left (b^{3} c^{2} - a^{2} b c\right )}^{2} a b^{3} \sqrt {c} d^{3} - {\left (3 \, b^{7} c^{3} - 4 \, a^{2} b^{5} c^{2} + a^{4} b^{3} c\right )} d^{3} {\left | b^{3} c^{2} - a^{2} b c \right |} + {\left (3 \, a b^{9} c^{\frac {9}{2}} - 8 \, a^{3} b^{7} c^{\frac {7}{2}} + 7 \, a^{5} b^{5} c^{\frac {5}{2}} - 2 \, a^{7} b^{3} c^{\frac {3}{2}}\right )} d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c^{2} - a^{3} c + \sqrt {{\left (a b^{2} c^{2} - a^{3} c\right )}^{2} + {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )} {\left (b^{2} c^{2} - a^{2} c\right )}}}{b^{2} c^{2} - a^{2} c}}}\right )}{{\left (b^{5} c^{\frac {9}{2}} - a b^{4} c^{4} - 2 \, a^{2} b^{3} c^{\frac {7}{2}} + 2 \, a^{3} b^{2} c^{3} + a^{4} b c^{\frac {5}{2}} - a^{5} c^{2}\right )} \sqrt {-b \sqrt {c} - a} {\left | b^{3} c^{2} - a^{2} b c \right |}} + \frac {{\left ({\left (b^{3} c^{2} - a^{2} b c\right )}^{2} a b^{3} d^{3} + {\left (3 \, b^{7} c^{\frac {5}{2}} - 4 \, a^{2} b^{5} c^{\frac {3}{2}} + a^{4} b^{3} \sqrt {c}\right )} d^{3} {\left | b^{3} c^{2} - a^{2} b c \right |} + {\left (3 \, a b^{9} c^{4} - 8 \, a^{3} b^{7} c^{3} + 7 \, a^{5} b^{5} c^{2} - 2 \, a^{7} b^{3} c\right )} d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c^{2} - a^{3} c - \sqrt {{\left (a b^{2} c^{2} - a^{3} c\right )}^{2} + {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )} {\left (b^{2} c^{2} - a^{2} c\right )}}}{b^{2} c^{2} - a^{2} c}}}\right )}{{\left (b^{5} c^{4} + a b^{4} c^{\frac {7}{2}} - 2 \, a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{\frac {5}{2}} + a^{4} b c^{2} + a^{5} c^{\frac {3}{2}}\right )} \sqrt {b \sqrt {c} - a} {\left | b^{3} c^{2} - a^{2} b c \right |}} - \frac {2 \, {\left (3 \, \sqrt {\sqrt {d x + c} b + a} b^{7} c^{2} d^{3} + {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{5} c d^{3} - {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{5} c d^{3} - 4 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{5} c d^{3} - {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a b^{3} d^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} b^{3} d^{3} - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} b^{3} d^{3} + \sqrt {\sqrt {d x + c} b + a} a^{4} b^{3} d^{3}\right )}}{{\left (b^{2} c^{2} - a^{2} 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 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 784, normalized size = 3.50 \begin {gather*} -\frac {\left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a \,b^{4} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2} \left (-b^{2} c +a^{2}\right )}+\frac {3 b^{4} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{16 \left (-b^{2} c +a^{2}\right ) \sqrt {b^{2} c}\, \sqrt {-a -\sqrt {b^{2} c}}}-\frac {3 b^{4} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{16 \left (-b^{2} c +a^{2}\right ) \sqrt {b^{2} c}\, \sqrt {-a +\sqrt {b^{2} c}}}-\frac {3 \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a^{3} b^{2} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2} \left (-b^{2} c +a^{2}\right ) c}-\frac {a^{2} b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{8 \left (-b^{2} c +a^{2}\right ) \sqrt {b^{2} c}\, \sqrt {-a -\sqrt {b^{2} c}}\, c}+\frac {a^{2} b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{8 \left (-b^{2} c +a^{2}\right ) \sqrt {b^{2} c}\, \sqrt {-a +\sqrt {b^{2} c}}\, c}+\frac {\left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}} b^{4} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2} \left (-b^{2} c +a^{2}\right )}-\frac {3 \sqrt {a +\sqrt {d x +c}\, b}\, b^{4} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2}}+\frac {3 \left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}} a^{2} b^{2} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2} \left (-b^{2} c +a^{2}\right ) c}+\frac {\sqrt {a +\sqrt {d x +c}\, b}\, a^{2} b^{2} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2} c}-\frac {a \,b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{16 \left (-b^{2} c +a^{2}\right ) \sqrt {-a -\sqrt {b^{2} c}}\, c}-\frac {a \,b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{16 \left (-b^{2} c +a^{2}\right ) \sqrt {-a +\sqrt {b^{2} c}}\, c}-\frac {\left (a +\sqrt {d x +c}\, b \right )^{\frac {7}{2}} a \,b^{2} d^{2}}{8 \left (-b^{2} c +\left (d x +c \right ) b^{2}\right )^{2} \left (-b^{2} c +a^{2}\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 {\sqrt {\sqrt {d x + c} b + a}}{x^{3}}\,{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 = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,\sqrt {c+d\,x}}}{x^3} \,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 + d x}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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