Integrand size = 17, antiderivative size = 110 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {5 d (b c-a d) \sqrt {c+d x}}{b^3}+\frac {5 d (c+d x)^{3/2}}{3 b^2}-\frac {(c+d x)^{5/2}}{b (a+b x)}-\frac {5 d (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 52, 65, 214} \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {5 d (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}}+\frac {5 d \sqrt {c+d x} (b c-a d)}{b^3}-\frac {(c+d x)^{5/2}}{b (a+b x)}+\frac {5 d (c+d x)^{3/2}}{3 b^2} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{5/2}}{b (a+b x)}+\frac {(5 d) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{2 b} \\ & = \frac {5 d (c+d x)^{3/2}}{3 b^2}-\frac {(c+d x)^{5/2}}{b (a+b x)}+\frac {(5 d (b c-a d)) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b^2} \\ & = \frac {5 d (b c-a d) \sqrt {c+d x}}{b^3}+\frac {5 d (c+d x)^{3/2}}{3 b^2}-\frac {(c+d x)^{5/2}}{b (a+b x)}+\frac {\left (5 d (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^3} \\ & = \frac {5 d (b c-a d) \sqrt {c+d x}}{b^3}+\frac {5 d (c+d x)^{3/2}}{3 b^2}-\frac {(c+d x)^{5/2}}{b (a+b x)}+\frac {\left (5 (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^3} \\ & = \frac {5 d (b c-a d) \sqrt {c+d x}}{b^3}+\frac {5 d (c+d x)^{3/2}}{3 b^2}-\frac {(c+d x)^{5/2}}{b (a+b x)}-\frac {5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {\sqrt {c+d x} \left (15 a^2 d^2+10 a b d (-2 c+d x)+b^2 \left (3 c^2-14 c d x-2 d^2 x^2\right )\right )}{3 b^3 (a+b x)}+\frac {5 d (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{7/2}} \]
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Time = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09
method | result | size |
risch | \(-\frac {2 d \left (-b d x +6 a d -7 b c \right ) \sqrt {d x +c}}{3 b^{3}}+\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) d \left (-\frac {\sqrt {d x +c}}{2 \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{3}}\) | \(120\) |
pseudoelliptic | \(-\frac {5 \left (-d \left (a d -b c \right )^{2} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (\frac {\left (-\frac {2}{3} d^{2} x^{2}-\frac {14}{3} c d x +c^{2}\right ) b^{2}}{5}-\frac {4 d a \left (-\frac {d x}{2}+c \right ) b}{3}+a^{2} d^{2}\right ) \sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3} \left (b x +a \right )}\) | \(127\) |
derivativedivides | \(2 d \left (-\frac {-\frac {b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \sqrt {d x +c}\, a d -2 \sqrt {d x +c}\, b c}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{3}}\right )\) | \(152\) |
default | \(2 d \left (-\frac {-\frac {b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \sqrt {d x +c}\, a d -2 \sqrt {d x +c}\, b c}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{3}}\right )\) | \(152\) |
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Time = 0.26 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.00 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\left [-\frac {15 \, {\left (a b c d - a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \, {\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{6 \, {\left (b^{4} x + a b^{3}\right )}}, -\frac {15 \, {\left (a b c d - a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \, {\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {5 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} - \frac {\sqrt {d x + c} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a b c d^{2} + \sqrt {d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{3}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{4} d + 6 \, \sqrt {d x + c} b^{4} c d - 6 \, \sqrt {d x + c} a b^{3} d^{2}\right )}}{3 \, b^{6}} \]
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Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {2\,d\,{\left (c+d\,x\right )}^{3/2}}{3\,b^2}-\frac {\sqrt {c+d\,x}\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{b^4\,\left (c+d\,x\right )-b^4\,c+a\,b^3\,d}+\frac {5\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,d\,{\left (a\,d-b\,c\right )}^{3/2}\,\sqrt {c+d\,x}}{a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d}\right )\,{\left (a\,d-b\,c\right )}^{3/2}}{b^{7/2}}+\frac {2\,d\,\left (2\,b^2\,c-2\,a\,b\,d\right )\,\sqrt {c+d\,x}}{b^4} \]
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