Optimal. Leaf size=114 \[ -\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}}+\frac {3 d \sqrt {c+d x}}{4 (a+b x) (b c-a d)^2}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 208} \[ -\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}}+\frac {3 d \sqrt {c+d x}}{4 (a+b x) (b c-a d)^2}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx &=-\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}-\frac {(3 d) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{4 (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}+\frac {\left (3 d^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 (b c-a d)^2}\\ &=-\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}+\frac {(3 d) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 (b c-a d)^2}\\ &=-\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}-\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.44 \[ \frac {2 d^2 \sqrt {c+d x} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {b (c+d x)}{a d-b c}\right )}{(a d-b c)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 549, normalized size = 4.82 \[ \left [\frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 148, normalized size = 1.30 \[ \frac {3 \, d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {3 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} - 5 \, \sqrt {d x + c} b c d^{2} + 5 \, \sqrt {d x + c} a d^{3}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 115, normalized size = 1.01 \[ \frac {3 d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {\sqrt {d x +c}\, d^{2}}{2 \left (a d -b c \right ) \left (b d x +a d \right )^{2}}+\frac {3 \sqrt {d x +c}\, d^{2}}{4 \left (a d -b c \right )^{2} \left (b d x +a d \right )} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 142, normalized size = 1.25 \[ \frac {\frac {5\,d^2\,\sqrt {c+d\,x}}{4\,\left (a\,d-b\,c\right )}+\frac {3\,b\,d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,{\left (a\,d-b\,c\right )}^2}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d}+\frac {3\,d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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