Optimal. Leaf size=261 \[ -\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}-\frac {1}{c \sqrt {\sqrt {a^2 x^2+b^2}+a x} (c x+d)} \]
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Rubi [A] time = 0.61, antiderivative size = 310, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2119, 1648, 12, 707, 1093, 205} \begin {gather*} -\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {2 a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{c \left (-c \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-2 a d \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2 c\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 707
Rule 1093
Rule 1648
Rule 2119
Rubi steps
\begin {align*} \int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=(2 a) \operatorname {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )^2} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c \left (b^2 c-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right )}+\frac {a \operatorname {Subst}\left (\int -\frac {2 b^2 \left (b^2 c^2+a^2 d^2\right )}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 b^2 c \left (b^2 c^2+a^2 d^2\right )}\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c \left (b^2 c-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right )}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c}\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c \left (b^2 c-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right )}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-b^2 c+2 a d x^2+c x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c \left (b^2 c-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right )}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a d-\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {b^2 c^2+a^2 d^2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a d+\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {b^2 c^2+a^2 d^2}}\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c \left (b^2 c-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right )}-\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 273, normalized size = 1.05 \begin {gather*} -\frac {a \left (-\frac {\sqrt {-\sqrt {a^2 d^2+b^2 c^2}-a d} \tan ^{-1}\left (\frac {b \sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x} \sqrt {-\sqrt {a^2 d^2+b^2 c^2}-a d}}\right )}{b \sqrt {a^2 d^2+b^2 c^2}}+\frac {\sqrt {\sqrt {a^2 d^2+b^2 c^2}-a d} \tan ^{-1}\left (\frac {b \sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x} \sqrt {\sqrt {a^2 d^2+b^2 c^2}-a d}}\right )}{b \sqrt {a^2 d^2+b^2 c^2}}+\frac {\sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x} (a c x+a d)}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.04, size = 261, normalized size = 1.00 \begin {gather*} -\frac {1}{c (d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 1174, normalized size = 4.50 \begin {gather*} \frac {{\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} + {\left (a b^{2} c^{2} + a^{3} d^{2} - {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) - {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} - {\left (a b^{2} c^{2} + a^{3} d^{2} - {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) + {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} + {\left (a b^{2} c^{2} + a^{3} d^{2} + {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) - {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} - {\left (a b^{2} c^{2} + a^{3} d^{2} + {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{2 \, {\left (b^{2} c^{2} x + b^{2} c d\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (c x + d\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c x +d \right )^{2} \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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 {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (c x + d\right )}^{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 = -0.00 \begin {gather*} \int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,{\left (d+c\,x\right )}^2} \,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}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x + d\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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