Optimal. Leaf size=86 \[ -\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}}+\frac {2 \sqrt {c+d x} (b c-a d)}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b} \]
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Rubi [A] time = 0.05, antiderivative size = 86, 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 = {50, 63, 208} \[ \frac {2 \sqrt {c+d x} (b c-a d)}{b^2}-\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}}+\frac {2 (c+d x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{a+b x} \, dx &=\frac {2 (c+d x)^{3/2}}{3 b}+\frac {(b c-a d) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b}\\ &=\frac {2 (b c-a d) \sqrt {c+d x}}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b}+\frac {(b c-a d)^2 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^2}\\ &=\frac {2 (b c-a d) \sqrt {c+d x}}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b}+\frac {\left (2 (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^2 d}\\ &=\frac {2 (b c-a d) \sqrt {c+d x}}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b}-\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 77, normalized size = 0.90 \[ \frac {2 \sqrt {c+d x} (-3 a d+4 b c+b d x)}{3 b^2}-\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 188, normalized size = 2.19 \[ \left [-\frac {3 \, {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left (b d x + 4 \, b c - 3 \, a d\right )} \sqrt {d x + c}}{3 \, b^{2}}, -\frac {2 \, {\left (3 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (b d x + 4 \, b c - 3 \, a d\right )} \sqrt {d x + c}\right )}}{3 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 105, normalized size = 1.22 \[ \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{2}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{2} + 3 \, \sqrt {d x + c} b^{2} c - 3 \, \sqrt {d x + c} a b d\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 167, normalized size = 1.94 \[ \frac {2 a^{2} d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{2}}-\frac {4 a c d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b}+\frac {2 c^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}-\frac {2 \sqrt {d x +c}\, a d}{b^{2}}+\frac {2 \sqrt {d x +c}\, c}{b}+\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 b} \]
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 = 0.07, size = 93, normalized size = 1.08 \[ \frac {2\,{\left (c+d\,x\right )}^{3/2}}{3\,b}-\frac {2\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{b^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\sqrt {c+d\,x}}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^{3/2}}{b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.70, size = 82, normalized size = 0.95 \[ \frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 b} + \frac {\sqrt {c + d x} \left (- 2 a d + 2 b c\right )}{b^{2}} + \frac {2 \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{3} \sqrt {\frac {a d - b c}{b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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