Integrand size = 17, antiderivative size = 112 \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\frac {2 (b c-a d)^2 \sqrt {c+d x}}{b^3}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac {2 (c+d x)^{5/2}}{5 b}-\frac {2 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 65, 214} \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=-\frac {2 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}}+\frac {2 \sqrt {c+d x} (b c-a d)^2}{b^3}+\frac {2 (c+d x)^{3/2} (b c-a d)}{3 b^2}+\frac {2 (c+d x)^{5/2}}{5 b} \]
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Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2 (c+d x)^{5/2}}{5 b}+\frac {(b c-a d) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b} \\ & = \frac {2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac {2 (c+d x)^{5/2}}{5 b}+\frac {(b c-a d)^2 \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^2} \\ & = \frac {2 (b c-a d)^2 \sqrt {c+d x}}{b^3}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac {2 (c+d x)^{5/2}}{5 b}+\frac {(b c-a d)^3 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^3} \\ & = \frac {2 (b c-a d)^2 \sqrt {c+d x}}{b^3}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac {2 (c+d x)^{5/2}}{5 b}+\frac {\left (2 (b c-a d)^3\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^3 d} \\ & = \frac {2 (b c-a d)^2 \sqrt {c+d x}}{b^3}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac {2 (c+d x)^{5/2}}{5 b}-\frac {2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {c+d x} \left (15 a^2 d^2-5 a b d (7 c+d x)+b^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )}{15 b^3}-\frac {2 (-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{7/2}} \]
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Time = 0.55 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\left (\frac {\left (d^{2} x^{2}+\frac {11}{3} c d x +\frac {23}{3} c^{2}\right ) b^{2}}{5}-\frac {7 \left (\frac {d x}{7}+c \right ) d a b}{3}+a^{2} d^{2}\right ) \sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3}}\) | \(115\) |
risch | \(\frac {2 \left (3 d^{2} x^{2} b^{2}-5 x a b \,d^{2}+11 x \,b^{2} c d +15 a^{2} d^{2}-35 a b c d +23 b^{2} c^{2}\right ) \sqrt {d x +c}}{15 b^{3}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{3} \sqrt {\left (a d -b c \right ) b}}\) | \(139\) |
derivativedivides | \(\frac {\frac {2 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {2 a b d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \sqrt {d x +c}\, a^{2} d^{2}-4 \sqrt {d x +c}\, a b c d +2 b^{2} c^{2} \sqrt {d x +c}}{b^{3}}+\frac {2 \left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{3} \sqrt {\left (a d -b c \right ) b}}\) | \(161\) |
default | \(\frac {\frac {2 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {2 a b d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \sqrt {d x +c}\, a^{2} d^{2}-4 \sqrt {d x +c}\, a b c d +2 b^{2} c^{2} \sqrt {d x +c}}{b^{3}}+\frac {2 \left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{3} \sqrt {\left (a d -b c \right ) b}}\) | \(161\) |
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Time = 0.24 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.59 \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\left [\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, b^{3}}, -\frac {2 \, {\left (15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (3 \, b^{2} d^{2} x^{2} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}\right )}}{15 \, b^{3}}\right ] \]
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Time = 1.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.30 \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {d \left (c + d x\right )^{\frac {5}{2}}}{5 b} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (- a d^{2} + b c d\right )}{3 b^{2}} + \frac {\sqrt {c + d x} \left (a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d\right )}{b^{3}} - \frac {d \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{4} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.53 \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} + 5 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c + 15 \, \sqrt {d x + c} b^{4} c^{2} - 5 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} d - 30 \, \sqrt {d x + c} a b^{3} c d + 15 \, \sqrt {d x + c} a^{2} b^{2} d^{2}\right )}}{15 \, b^{5}} \]
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Time = 0.48 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x)^{5/2}}{a+b x} \, dx=\frac {2\,{\left (c+d\,x\right )}^{5/2}}{5\,b}-\frac {2\,\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b^2}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\sqrt {c+d\,x}}{b^3}-\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{7/2}} \]
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