Optimal. Leaf size=130 \[ -\frac {a-b \sqrt {c+d x}}{x \left (a^2-b^2 c\right )}+\frac {a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac {2 a b^2 d \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac {b d \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2-b^2 c\right )^2} \]
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Rubi [A] time = 0.18, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {371, 1398, 823, 801, 635, 206, 260} \begin {gather*} -\frac {a-b \sqrt {c+d x}}{x \left (a^2-b^2 c\right )}+\frac {a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac {2 a b^2 d \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac {b d \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2-b^2 c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 260
Rule 371
Rule 635
Rule 801
Rule 823
Rule 1398
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \sqrt {c+d x}\right )} \, dx &=d \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right ) (-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \operatorname {Subst}\left (\int \frac {x}{(a+b x) \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {a-b \sqrt {c+d x}}{\left (a^2-b^2 c\right ) x}+\frac {d \operatorname {Subst}\left (\int \frac {-a b c+b^2 c x}{(a+b x) \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{c \left (a^2-b^2 c\right )}\\ &=-\frac {a-b \sqrt {c+d x}}{\left (a^2-b^2 c\right ) x}+\frac {d \operatorname {Subst}\left (\int \left (-\frac {2 a b^3 c}{\left (a^2-b^2 c\right ) (a+b x)}-\frac {b c \left (a^2+b^2 c-2 a b x\right )}{\left (-a^2+b^2 c\right ) \left (c-x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{c \left (a^2-b^2 c\right )}\\ &=-\frac {a-b \sqrt {c+d x}}{\left (a^2-b^2 c\right ) x}-\frac {2 a b^2 d \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac {(b d) \operatorname {Subst}\left (\int \frac {a^2+b^2 c-2 a b x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}\\ &=-\frac {a-b \sqrt {c+d x}}{\left (a^2-b^2 c\right ) x}-\frac {2 a b^2 d \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}-\frac {\left (2 a b^2 d\right ) \operatorname {Subst}\left (\int \frac {x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac {\left (b \left (a^2+b^2 c\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}\\ &=-\frac {a-b \sqrt {c+d x}}{\left (a^2-b^2 c\right ) x}+\frac {b \left (a^2+b^2 c\right ) d \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2-b^2 c\right )^2}+\frac {a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac {2 a b^2 d \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 144, normalized size = 1.11 \begin {gather*} \frac {\sqrt {c} \left (-\left (a^2-b^2 c\right ) \left (a-b \sqrt {c+d x}\right )-a b^2 d x \log \left (a^2-b^2 (c+d x)\right )+a b^2 d x \log (x)\right )+b d x \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )-2 a b^2 \sqrt {c} d x \tanh ^{-1}\left (\frac {b \sqrt {c+d x}}{a}\right )}{\sqrt {c} x \left (a^2-b^2 c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 135, normalized size = 1.04 \begin {gather*} -\frac {a-b \sqrt {c+d x}}{x \left (a^2-b^2 c\right )}+\frac {a b^2 d \log (-d x)}{\left (a^2-b^2 c\right )^2}-\frac {2 a b^2 d \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac {d \left (a^2 b+b^3 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c} \left (b^2 c-a^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 283, normalized size = 2.18 \begin {gather*} \left [-\frac {4 \, a b^{2} c d x \log \left (\sqrt {d x + c} b + a\right ) - 2 \, a b^{2} c d x \log \relax (x) - 2 \, a b^{2} c^{2} - {\left (b^{3} c + a^{2} b\right )} \sqrt {c} d x \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, a^{3} c + 2 \, {\left (b^{3} c^{2} - a^{2} b c\right )} \sqrt {d x + c}}{2 \, {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )} x}, -\frac {2 \, a b^{2} c d x \log \left (\sqrt {d x + c} b + a\right ) - a b^{2} c d x \log \relax (x) - a b^{2} c^{2} + {\left (b^{3} c + a^{2} b\right )} \sqrt {-c} d x \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + a^{3} c + {\left (b^{3} c^{2} - a^{2} b c\right )} \sqrt {d x + c}}{{\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 191, normalized size = 1.47 \begin {gather*} -\frac {2 \, a b^{3} d \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b} + \frac {a b^{2} d \log \left (-d x\right )}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}} - \frac {{\left (b^{3} c d + a^{2} b d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt {-c}} + \frac {a b^{2} c d - a^{3} d - {\left (b^{3} c d - a^{2} b d\right )} \sqrt {d x + c}}{{\left (b^{2} c - a^{2}\right )}^{2} d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 216, normalized size = 1.66 \begin {gather*} \frac {b^{3} \sqrt {c}\, d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\left (-b^{2} c +a^{2}\right )^{2}}+\frac {a \,b^{2} d \ln \left (d x \right )}{\left (-b^{2} c +a^{2}\right )^{2}}-\frac {2 a \,b^{2} d \ln \left (a +\sqrt {d x +c}\, b \right )}{\left (-b^{2} c +a^{2}\right )^{2}}+\frac {a^{2} b d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\left (-b^{2} c +a^{2}\right )^{2} \sqrt {c}}+\frac {a \,b^{2} c}{\left (-b^{2} c +a^{2}\right )^{2} x}-\frac {\sqrt {d x +c}\, b^{3} c}{\left (-b^{2} c +a^{2}\right )^{2} x}-\frac {a^{3}}{\left (-b^{2} c +a^{2}\right )^{2} x}+\frac {\sqrt {d x +c}\, a^{2} b}{\left (-b^{2} c +a^{2}\right )^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 191, normalized size = 1.47 \begin {gather*} \frac {1}{2} \, {\left (\frac {2 \, a b^{2} \log \left (d x\right )}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}} - \frac {4 \, a b^{2} \log \left (\sqrt {d x + c} b + a\right )}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}} - \frac {{\left (b^{3} c + a^{2} b\right )} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt {c}} + \frac {2 \, {\left (\sqrt {d x + c} b - a\right )}}{b^{2} c^{2} - a^{2} c - {\left (b^{2} c - a^{2}\right )} {\left (d x + c\right )}}\right )} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 220, normalized size = 1.69 \begin {gather*} \frac {\ln \left (\sqrt {c+d\,x}-\sqrt {c}\right )\,\left (4\,a\,b^2\,c\,d-b\,\sqrt {c}\,d\,\left (2\,a^2+2\,c\,b^2\right )\right )}{4\,a^4\,c-8\,a^2\,b^2\,c^2+4\,b^4\,c^3}+\frac {\ln \left (\sqrt {c+d\,x}+\sqrt {c}\right )\,\left (4\,a\,b^2\,c\,d+b\,\sqrt {c}\,d\,\left (2\,a^2+2\,c\,b^2\right )\right )}{4\,a^4\,c-8\,a^2\,b^2\,c^2+4\,b^4\,c^3}+\frac {\frac {a\,d}{b^2\,c-a^2}-\frac {b\,d\,\sqrt {c+d\,x}}{b^2\,c-a^2}}{d\,x}-\frac {2\,a\,b^2\,d\,\ln \left (a+b\,\sqrt {c+d\,x}\right )}{{\left (b^2\,c-a^2\right )}^2} \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} \left (a + b \sqrt {c + d x}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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