Optimal. Leaf size=86 \[ \frac {2 (b c-a d) \sqrt {c+d x}}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b}-\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.03, 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 = {52, 65, 214}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
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 ) \text {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 = 0.14, size = 77, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {c+d x} (4 b c-3 a d+b d x)}{3 b^2}+\frac {2 (-b c+a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 8.94, size = 86, normalized size = 1.00 \begin {gather*} \frac {-2 a d \sqrt {c+d x}}{b^2}+\frac {2 \left (c+d x\right )^{\frac {3}{2}}}{3 b}+\frac {2 b \text {ArcTan}\left [\frac {\sqrt {c+d x}}{\sqrt {\frac {a d}{b}-c}}\right ] \left (\frac {a d}{b}-c\right )^{\frac {7}{2}}}{\left (a d-b c\right )^2}+\frac {2 c \sqrt {c+d x}}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 99, normalized size = 1.15
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {b \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \sqrt {d x +c}-b c \sqrt {d x +c}\right )}{b^{2}}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(99\) |
default | \(-\frac {2 \left (-\frac {b \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \sqrt {d x +c}-b c \sqrt {d x +c}\right )}{b^{2}}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(99\) |
risch | \(-\frac {2 \left (-b d x +3 a d -4 b c \right ) \sqrt {d x +c}}{3 b^{2}}+\frac {2 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a^{2} d^{2}}{b^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {4 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a c d}{b \sqrt {\left (a d -b c \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{2}}{\sqrt {\left (a d -b c \right ) b}}\) | \(154\) |
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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Fricas [A]
time = 0.31, size = 188, normalized size = 2.19 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.89, size = 82, normalized size = 0.95 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 133, normalized size = 1.55 \begin {gather*} \frac {\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) b^{2}-2 \sqrt {c+d x} d b a+2 \sqrt {c+d x} c b^{2}}{b^{3}}+\frac {\left (4 d^{2} a^{2}-8 d c b a+4 c^{2} b^{2}\right ) \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{2 b^{2} \sqrt {-b^{2} c+a b d}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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