Optimal. Leaf size=140 \[ -\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)} \]
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Rubi [A] time = 0.16, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 151, 156, 63, 208} \begin {gather*} -\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {c+d x}}{a x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 99
Rule 151
Rule 156
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx &=-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {\int \frac {\frac {1}{2} (-4 b c+a d)-\frac {3 b d x}{2}}{x (a+b x)^2 \sqrt {c+d x}} \, dx}{a}\\ &=-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {\int \frac {-\frac {1}{2} (b c-a d) (4 b c-a d)-b d (b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a^2 (b c-a d)}\\ &=-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(b (4 b c-3 a d)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^3}-\frac {(4 b c-a d) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 a^3}\\ &=-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(b (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d}-\frac {(4 b c-a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d}\\ &=-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 169, normalized size = 1.21 \begin {gather*} \frac {\sqrt {c} \left (a (a+2 b x) \sqrt {c+d x} (b c-a d)+\sqrt {b} x (a+b x) (4 b c-3 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )\right )-x (a+b x) \left (a^2 d^2-5 a b c d+4 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c} x (a+b x) (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.70, size = 157, normalized size = 1.12 \begin {gather*} \frac {\left (3 a \sqrt {b} d-4 b^{3/2} c\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{a^3 \sqrt {a d-b c}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {c+d x} (a d+2 b (c+d x)-2 b c)}{a^2 x (a d+b (c+d x)-b c)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.87, size = 798, normalized size = 5.70 \begin {gather*} \left [-\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{a^{3} b c x^{2} + a^{4} c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.37, size = 166, normalized size = 1.19 \begin {gather*} \frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x + c} b c d + \sqrt {d x + c} a d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 167, normalized size = 1.19 \begin {gather*} -\frac {3 b d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{2}}+\frac {4 b^{2} c \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{3}}-\frac {\sqrt {d x +c}\, b d}{\left (b d x +a d \right ) a^{2}}-\frac {d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} \sqrt {c}}+\frac {4 b \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3}}-\frac {\sqrt {d x +c}}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 1175, normalized size = 8.39
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 68.83, size = 790, normalized size = 5.64 \begin {gather*} \frac {2 b^{2} c d \sqrt {c + d x}}{2 a^{4} d^{2} - 2 a^{3} b c d + 2 a^{3} b d^{2} x - 2 a^{2} b^{2} c d x} - \frac {2 b d^{2} \sqrt {c + d x}}{2 a^{3} d^{2} - 2 a^{2} b c d + 2 a^{2} b d^{2} x - 2 a b^{2} c d x} + \frac {b d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {b d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {b^{2} c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} + \frac {b^{2} c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} - \frac {c d \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} + \frac {c d \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} - \frac {2 d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{2} \sqrt {- c}} - \frac {\sqrt {c + d x}}{a^{2} x} + \frac {4 b c \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{3} \sqrt {\frac {a d}{b} - c}} - \frac {4 b c \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{3} \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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