Integrand size = 32, antiderivative size = 253 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\frac {a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+3 A d^3-2 c^3 D\right )}{b^3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {2 D \sqrt {c+d x}}{b^2 d^2}-\frac {\left (b^3 (2 B c-3 A d)-a b^2 (4 c C-B d)-3 a^3 d D+a^2 b (C d+6 c D)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{5/2}} \]
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Time = 0.39 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1635, 911, 1275, 214} \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=-\frac {A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}}{(a+b x) \sqrt {c+d x} (b c-a d)}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (-3 a^3 d D+a^2 b (6 c D+C d)-a b^2 (4 c C-B d)+b^3 (2 B c-3 A d)\right )}{b^{5/2} (b c-a d)^{5/2}}+\frac {a^3 d^3 D-a^2 b C d^3+a b^2 B d^3-\left (b^3 \left (3 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right )\right )}{b^3 d^2 \sqrt {c+d x} (b c-a d)^2}+\frac {2 D \sqrt {c+d x}}{b^2 d^2} \]
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Rule 214
Rule 911
Rule 1275
Rule 1635
Rubi steps \begin{align*} \text {integral}& = -\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {\int \frac {-\frac {b^3 (2 B c-3 A d)-a b^2 (2 c C-B d)+a^3 d D-a^2 b (C d-2 c D)}{2 b^3}-\frac {(b c-a d) (b C-a D) x}{b^2}-\left (c-\frac {a d}{b}\right ) D x^2}{(a+b x) (c+d x)^{3/2}} \, dx}{-b c+a d} \\ & = -\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) \sqrt {c+d x}}-\frac {2 \text {Subst}\left (\int \frac {\frac {-c^2 \left (c-\frac {a d}{b}\right ) D+\frac {c d (b c-a d) (b C-a D)}{b^2}-\frac {d^2 \left (b^3 (2 B c-3 A d)-a b^2 (2 c C-B d)+a^3 d D-a^2 b (C d-2 c D)\right )}{2 b^3}}{d^2}-\frac {\left (-2 c \left (c-\frac {a d}{b}\right ) D+\frac {d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac {\left (c-\frac {a d}{b}\right ) D x^4}{d^2}}{x^2 \left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )} \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)} \\ & = -\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) \sqrt {c+d x}}-\frac {2 \text {Subst}\left (\int \left (-\frac {(b c-a d) D}{b^2 d}+\frac {a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+3 A d^3-2 c^3 D\right )}{2 b^3 d (b c-a d) x^2}+\frac {d \left (b^3 (2 B c-3 A d)-a b^2 (4 c C-B d)-3 a^3 d D+a^2 b (C d+6 c D)\right )}{2 b^2 (b c-a d) \left (b c-a d-b x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)} \\ & = \frac {a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+3 A d^3-2 c^3 D\right )}{b^3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {2 D \sqrt {c+d x}}{b^2 d^2}-\frac {\left (b^3 (2 B c-3 A d)-a b^2 (4 c C-B d)-3 a^3 d D+a^2 b (C d+6 c D)\right ) \text {Subst}\left (\int \frac {1}{b c-a d-b x^2} \, dx,x,\sqrt {c+d x}\right )}{b^2 (b c-a d)^2} \\ & = \frac {a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+3 A d^3-2 c^3 D\right )}{b^3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {2 D \sqrt {c+d x}}{b^2 d^2}-\frac {\left (b^3 (2 B c-3 A d)-a b^2 (4 c C-B d)-3 a^3 d D+a^2 b (C d+6 c D)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\frac {3 a^3 d^2 D (c+d x)+a^2 b d (c+d x) (-C d-4 c D+2 d D x)+b^3 \left (-A d^2 (c+3 d x)+2 c x \left (-c C d+B d^2+2 c^2 D+c d D x\right )\right )+a b^2 \left (4 c^3 D+d^3 (-2 A+B x)-2 c^2 d (C+D x)+c d^2 \left (3 B-4 D x^2\right )\right )}{b^2 d^2 (b c-a d)^2 (a+b x) \sqrt {c+d x}}+\frac {\left (b^3 (2 B c-3 A d)+a b^2 (-4 c C+B d)-3 a^3 d D+a^2 b (C d+6 c D)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{5/2} (-b c+a d)^{5/2}} \]
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Time = 1.83 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {2 D \sqrt {d x +c}}{b^{2}}-\frac {2 d^{2} \left (\frac {\left (\frac {1}{2} A \,b^{3} d -\frac {1}{2} B a \,b^{2} d +\frac {1}{2} C \,a^{2} b d -\frac {1}{2} a^{3} d D\right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (3 A \,b^{3} d -B a \,b^{2} d -2 B \,b^{3} c -C \,a^{2} b d +4 C a \,b^{2} c +3 a^{3} d D-6 D a^{2} b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} b^{2}}-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{\left (a d -b c \right )^{2} \sqrt {d x +c}}}{d^{2}}\) | \(228\) |
default | \(\frac {\frac {2 D \sqrt {d x +c}}{b^{2}}-\frac {2 d^{2} \left (\frac {\left (\frac {1}{2} A \,b^{3} d -\frac {1}{2} B a \,b^{2} d +\frac {1}{2} C \,a^{2} b d -\frac {1}{2} a^{3} d D\right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (3 A \,b^{3} d -B a \,b^{2} d -2 B \,b^{3} c -C \,a^{2} b d +4 C a \,b^{2} c +3 a^{3} d D-6 D a^{2} b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} b^{2}}-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{\left (a d -b c \right )^{2} \sqrt {d x +c}}}{d^{2}}\) | \(228\) |
pseudoelliptic | \(-\frac {3 \left (\left (\left (b^{3} A -\frac {1}{3} a \,b^{2} B -\frac {1}{3} C \,a^{2} b +D a^{3}\right ) d -\frac {2 b c \left (B \,b^{2}-2 C a b +3 D a^{2}\right )}{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) \left (b x +a \right ) d^{2} \sqrt {d x +c}+\frac {2 \sqrt {\left (a d -b c \right ) b}\, \left (\left (\frac {3 A \,b^{3} x}{2}+\left (-\frac {B x}{2}+A \right ) a \,b^{2}+\frac {a^{2} x \left (-2 D x +C \right ) b}{2}-\frac {3 D a^{3} x}{2}\right ) d^{3}+\frac {c \left (\left (-2 B x +A \right ) b^{3}-3 a \left (-\frac {4 D x^{2}}{3}+B \right ) b^{2}+a^{2} \left (2 D x +C \right ) b -3 D a^{3}\right ) d^{2}}{2}+\left (\left (-D x +C \right ) b +2 D a \right ) b \,c^{2} \left (b x +a \right ) d -2 D b^{2} c^{3} \left (b x +a \right )\right )}{3}\right )}{\sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\, d^{2} b^{2} \left (b x +a \right ) \left (a d -b c \right )^{2}}\) | \(277\) |
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Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (236) = 472\).
Time = 0.33 (sec) , antiderivative size = 1583, normalized size of antiderivative = 6.26 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.53 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\frac {{\left (6 \, D a^{2} b c - 4 \, C a b^{2} c + 2 \, B b^{3} c - 3 \, D a^{3} d + C a^{2} b d + B a b^{2} d - 3 \, A b^{3} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (d x + c\right )} D b^{3} c^{3} - 2 \, D b^{3} c^{4} - 2 \, {\left (d x + c\right )} C b^{3} c^{2} d + 2 \, D a b^{2} c^{3} d + 2 \, C b^{3} c^{3} d + 2 \, {\left (d x + c\right )} B b^{3} c d^{2} - 2 \, C a b^{2} c^{2} d^{2} - 2 \, B b^{3} c^{2} d^{2} + {\left (d x + c\right )} D a^{3} d^{3} - {\left (d x + c\right )} C a^{2} b d^{3} + {\left (d x + c\right )} B a b^{2} d^{3} - 3 \, {\left (d x + c\right )} A b^{3} d^{3} + 2 \, B a b^{2} c d^{3} + 2 \, A b^{3} c d^{3} - 2 \, A a b^{2} d^{4}}{{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} {\left ({\left (d x + c\right )}^{\frac {3}{2}} b - \sqrt {d x + c} b c + \sqrt {d x + c} a d\right )}} + \frac {2 \, \sqrt {d x + c} D}{b^{2} d^{2}} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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