Integrand size = 20, antiderivative size = 204 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}+\frac {a (4 b c-9 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}} \]
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Time = 0.15 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 81, 52, 65, 214} \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {a^2 (c+d x)^{7/2}}{b^2 (a+b x) (b c-a d)}+\frac {a (4 b c-9 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}-\frac {a \sqrt {c+d x} (4 b c-9 a d) (b c-a d)}{b^5}-\frac {a (c+d x)^{3/2} (4 b c-9 a d)}{3 b^4}-\frac {a (c+d x)^{5/2} (4 b c-9 a d)}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}+\frac {\int \frac {(c+d x)^{5/2} \left (-\frac {1}{2} a (2 b c-7 a d)+b (b c-a d) x\right )}{a+b x} \, dx}{b^2 (b c-a d)} \\ & = \frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {(a (4 b c-9 a d)) \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{2 b^2 (b c-a d)} \\ & = -\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {(a (4 b c-9 a d)) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{2 b^3} \\ & = -\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {(a (4 b c-9 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b^4} \\ & = -\frac {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {\left (a (4 b c-9 a d) (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^5} \\ & = -\frac {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {\left (a (4 b c-9 a d) (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^5 d} \\ & = -\frac {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}+\frac {a (4 b c-9 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {\sqrt {c+d x} \left (-945 a^4 d^3+210 a^3 b d^2 (8 c-3 d x)+30 b^4 x (c+d x)^3+7 a^2 b^2 d \left (-107 c^2+166 c d x+18 d^2 x^2\right )+2 a b^3 \left (15 c^3-277 c^2 d x-109 c d^2 x^2-27 d^3 x^3\right )\right )}{105 b^5 d (a+b x)}-\frac {a (4 b c-9 a d) (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{11/2}} \]
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Time = 0.64 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {9 \left (a d -b c \right )^{2} \left (a d -\frac {4 b c}{9}\right ) d \left (b x +a \right ) a \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-9 \sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\, \left (-\frac {2 x \left (d x +c \right )^{3} b^{4}}{63}-\frac {2 \left (-\frac {9}{5} d^{3} x^{3}-\frac {109}{15} c \,d^{2} x^{2}-\frac {277}{15} c^{2} d x +c^{3}\right ) a \,b^{3}}{63}+\frac {107 \left (-\frac {18}{107} d^{2} x^{2}-\frac {166}{107} c d x +c^{2}\right ) d \,a^{2} b^{2}}{135}-\frac {16 \left (-\frac {3 d x}{8}+c \right ) d^{2} a^{3} b}{9}+a^{4} d^{3}\right )}{b^{5} d \left (b x +a \right ) \sqrt {\left (a d -b c \right ) b}}\) | \(194\) |
risch | \(-\frac {2 \left (-15 d^{3} x^{3} b^{3}+42 x^{2} a \,b^{2} d^{3}-45 x^{2} b^{3} c \,d^{2}-105 x \,a^{2} b \,d^{3}+154 x a \,b^{2} c \,d^{2}-45 x \,b^{3} c^{2} d +420 a^{3} d^{3}-735 a^{2} b c \,d^{2}+322 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {d x +c}}{105 d \,b^{5}}+\frac {a \left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (-\frac {a d \sqrt {d x +c}}{2 \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (9 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{5}}\) | \(222\) |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a d \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-a^{2} b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+\frac {2 a \,b^{2} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+4 \sqrt {d x +c}\, a^{3} d^{3}-6 \sqrt {d x +c}\, a^{2} b c \,d^{2}+2 \sqrt {d x +c}\, a \,b^{2} c^{2} d \right )}{b^{5}}+\frac {2 d a \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (9 a^{3} d^{3}-22 a^{2} b c \,d^{2}+17 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{5}}}{d}\) | \(252\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a d \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-a^{2} b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+\frac {2 a \,b^{2} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+4 \sqrt {d x +c}\, a^{3} d^{3}-6 \sqrt {d x +c}\, a^{2} b c \,d^{2}+2 \sqrt {d x +c}\, a \,b^{2} c^{2} d \right )}{b^{5}}+\frac {2 d a \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (9 a^{3} d^{3}-22 a^{2} b c \,d^{2}+17 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{5}}}{d}\) | \(252\) |
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Time = 0.26 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.98 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\left [\frac {105 \, {\left (4 \, a^{2} b^{2} c^{2} d - 13 \, a^{3} b c d^{2} + 9 \, a^{4} d^{3} + {\left (4 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} + 9 \, a^{3} b d^{3}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (30 \, b^{4} d^{3} x^{4} + 30 \, a b^{3} c^{3} - 749 \, a^{2} b^{2} c^{2} d + 1680 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 18 \, {\left (5 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{3} + 2 \, {\left (45 \, b^{4} c^{2} d - 109 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (15 \, b^{4} c^{3} - 277 \, a b^{3} c^{2} d + 581 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x + c}}{210 \, {\left (b^{6} d x + a b^{5} d\right )}}, \frac {105 \, {\left (4 \, a^{2} b^{2} c^{2} d - 13 \, a^{3} b c d^{2} + 9 \, a^{4} d^{3} + {\left (4 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} + 9 \, a^{3} b d^{3}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (30 \, b^{4} d^{3} x^{4} + 30 \, a b^{3} c^{3} - 749 \, a^{2} b^{2} c^{2} d + 1680 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 18 \, {\left (5 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{3} + 2 \, {\left (45 \, b^{4} c^{2} d - 109 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (15 \, b^{4} c^{3} - 277 \, a b^{3} c^{2} d + 581 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x + c}}{105 \, {\left (b^{6} d x + a b^{5} d\right )}}\right ] \]
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Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.40 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {{\left (4 \, a b^{3} c^{3} - 17 \, a^{2} b^{2} c^{2} d + 22 \, a^{3} b c d^{2} - 9 \, a^{4} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{5}} - \frac {\sqrt {d x + c} a^{2} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a^{3} b c d^{2} + \sqrt {d x + c} a^{4} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{12} d^{6} - 42 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{11} d^{7} - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{11} c d^{7} - 210 \, \sqrt {d x + c} a b^{11} c^{2} d^{7} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{10} d^{8} + 630 \, \sqrt {d x + c} a^{2} b^{10} c d^{8} - 420 \, \sqrt {d x + c} a^{3} b^{9} d^{9}\right )}}{105 \, b^{14} d^{7}} \]
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Time = 0.48 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.05 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\left (\frac {\left (\frac {4\,c}{b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d}\right )\,{\left (a\,d-b\,c\right )}^2}{b^2}-\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {2\,c^2}{b^2\,d}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d}+\frac {2\,\left (\frac {4\,c}{b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b}\right )\,\sqrt {c+d\,x}+{\left (c+d\,x\right )}^{3/2}\,\left (\frac {2\,c^2}{3\,b^2\,d}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{3\,b^4\,d}+\frac {2\,\left (\frac {4\,c}{b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d}\right )\,\left (a\,d-b\,c\right )}{3\,b}\right )-\left (\frac {4\,c}{5\,b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{5\,b^3\,d}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,b^2\,d}-\frac {\sqrt {c+d\,x}\,\left (a^4\,d^3-2\,a^3\,b\,c\,d^2+a^2\,b^2\,c^2\,d\right )}{b^6\,\left (c+d\,x\right )-b^6\,c+a\,b^5\,d}+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (9\,a\,d-4\,b\,c\right )\,\sqrt {c+d\,x}}{9\,a^4\,d^3-22\,a^3\,b\,c\,d^2+17\,a^2\,b^2\,c^2\,d-4\,a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (9\,a\,d-4\,b\,c\right )}{b^{11/2}} \]
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