Integrand size = 32, antiderivative size = 438 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=-\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )}{6 b^3 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac {a^2 b C d^2+b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )-a^3 d^2 D+a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )}{b^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt {c+d x}}{4 b (b c-a d)^4 (a+b x)}-\frac {\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{9/2}} \]
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Time = 0.82 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1635, 911, 1273, 1275, 214} \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (-5 B d^2-8 c^2 D+8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{4 b^{3/2} (b c-a d)^{9/2}}+\frac {a^3 \left (-d^2\right ) D+a^2 b C d^2+a b^2 \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (7 A d^2-4 B c d+2 c^2 C\right )}{b^2 \sqrt {c+d x} (b c-a d)^4}-\frac {3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3-\left (b^3 \left (7 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right )\right )}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac {\sqrt {c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{4 b (a+b x) (b c-a d)^4} \]
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Rule 214
Rule 911
Rule 1273
Rule 1275
Rule 1635
Rubi steps \begin{align*} \text {integral}& = -\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac {\int \frac {-\frac {b^3 (4 B c-7 A d)-a b^2 (4 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-4 c D)}{2 b^3}-\frac {2 (b c-a d) (b C-a D) x}{b^2}-2 \left (c-\frac {a d}{b}\right ) D x^2}{(a+b x)^2 (c+d x)^{5/2}} \, dx}{2 (b c-a d)} \\ & = -\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac {\text {Subst}\left (\int \frac {\frac {-2 c^2 \left (c-\frac {a d}{b}\right ) D+\frac {2 c d (b c-a d) (b C-a D)}{b^2}-\frac {d^2 \left (b^3 (4 B c-7 A d)-a b^2 (4 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-4 c D)\right )}{2 b^3}}{d^2}-\frac {\left (-4 c \left (c-\frac {a d}{b}\right ) D+\frac {2 d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac {2 \left (c-\frac {a d}{b}\right ) D x^4}{d^2}}{x^4 \left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )^2} \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)} \\ & = -\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac {\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt {c+d x}}{4 b (b c-a d)^4 (a+b x)}+\frac {d^4 \text {Subst}\left (\int \frac {-\frac {(b c-a d)^2 \left (3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )\right )}{b d^6}-\frac {(b c-a d) \left (a^2 b C d^3-a^3 d^3 D-a b^2 d \left (8 c C d-3 B d^2-12 c^2 D\right )+b^3 \left (4 B c d^2-7 A d^3-4 c^3 D\right )\right ) x^2}{d^6}-\frac {b \left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) x^4}{2 d^4}}{x^4 \left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )} \, dx,x,\sqrt {c+d x}\right )}{2 b^2 (b c-a d)^4} \\ & = -\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac {\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt {c+d x}}{4 b (b c-a d)^4 (a+b x)}+\frac {d^4 \text {Subst}\left (\int \left (\frac {(b c-a d) \left (3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )\right )}{b d^5 x^4}+\frac {2 \left (-a^2 b C d^2-b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )+a^3 d^2 D-a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )\right )}{d^4 x^2}+\frac {b \left (-b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )-a^3 d^2 D-3 a^2 b d (C d-4 c D)-3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right )}{2 d^4 \left (b c-a d-b x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{2 b^2 (b c-a d)^4} \\ & = -\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )}{6 b^3 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac {a^2 b C d^2+b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )-a^3 d^2 D+a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )}{b^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt {c+d x}}{4 b (b c-a d)^4 (a+b x)}-\frac {\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right ) \text {Subst}\left (\int \frac {1}{b c-a d-b x^2} \, dx,x,\sqrt {c+d x}\right )}{4 b (b c-a d)^4} \\ & = -\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )}{6 b^3 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac {a^2 b C d^2+b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )-a^3 d^2 D+a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )}{b^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt {c+d x}}{4 b (b c-a d)^4 (a+b x)}-\frac {\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{9/2}} \\ \end{align*}
Time = 1.54 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {-3 a^4 d^2 D (c+d x)^2-4 b^4 c x \left (2 c x \left (-4 c C d+c^2 D-3 C d^2 x\right )+B d \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )+a^3 b d \left (-94 c^3 D+c^2 d (55 C-129 D x)+3 d^3 x \left (-8 B+5 C x+D x^2\right )-2 c d^2 \left (8 B-39 C x+12 D x^2\right )\right )-a^2 b^2 \left (8 c^4 D+3 d^4 x^2 (25 B-3 C x)+c^3 (-50 C d+164 d D x)+2 c d^3 x \left (67 B-66 C x+18 D x^2\right )+c^2 d^2 \left (83 B-149 C x+216 D x^2\right )\right )+A b d \left (-8 a^3 d^3+8 a^2 b d^2 (10 c+7 d x)+a b^2 d \left (39 c^2+238 c d x+175 d^2 x^2\right )+b^3 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )-a b^3 \left (B d \left (6 c^3+145 c^2 d x+160 c d^2 x^2+45 d^3 x^3\right )+8 c x \left (2 c^3 D-9 C d^3 x^2+c d^2 x (-17 C+9 D x)+c^2 (-11 C d+8 d D x)\right )\right )}{12 b d (b c-a d)^4 (a+b x)^2 (c+d x)^{3/2}}+\frac {\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)-3 a b^2 \left (-8 c C d+5 B d^2+8 c^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{4 b^{3/2} (-b c+a d)^{9/2}} \]
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Time = 1.96 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {\frac {2 d \left (\frac {\left (\frac {11}{8} A \,b^{3} d^{2}-\frac {7}{8} B a \,b^{2} d^{2}-\frac {1}{2} B \,b^{3} c d +\frac {3}{8} a^{2} b C \,d^{2}+C a \,b^{2} c d +\frac {1}{8} a^{3} d^{2} D-\frac {3}{2} D a^{2} b c d \right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {d \left (13 A a \,b^{3} d^{2}-13 A \,b^{4} c d -9 B \,a^{2} b^{2} d^{2}+5 B a \,b^{3} c d +4 B \,b^{4} c^{2}+5 C \,a^{3} b \,d^{2}+3 C \,a^{2} b^{2} c d -8 C a \,b^{3} c^{2}-D a^{4} d^{2}-11 D a^{3} b c d +12 D a^{2} b^{2} c^{2}\right ) \sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\left (35 A \,b^{3} d^{2}-15 B a \,b^{2} d^{2}-20 B \,b^{3} c d +3 a^{2} b C \,d^{2}+24 C a \,b^{2} c d +8 C \,b^{3} c^{2}+a^{3} d^{2} D-12 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 d \left (3 A b \,d^{2}-B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-3 D a \,c^{2}\right )}{\left (a d -b c \right )^{4} \sqrt {d x +c}}}{d}\) | \(459\) |
default | \(\frac {\frac {2 d \left (\frac {\left (\frac {11}{8} A \,b^{3} d^{2}-\frac {7}{8} B a \,b^{2} d^{2}-\frac {1}{2} B \,b^{3} c d +\frac {3}{8} a^{2} b C \,d^{2}+C a \,b^{2} c d +\frac {1}{8} a^{3} d^{2} D-\frac {3}{2} D a^{2} b c d \right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {d \left (13 A a \,b^{3} d^{2}-13 A \,b^{4} c d -9 B \,a^{2} b^{2} d^{2}+5 B a \,b^{3} c d +4 B \,b^{4} c^{2}+5 C \,a^{3} b \,d^{2}+3 C \,a^{2} b^{2} c d -8 C a \,b^{3} c^{2}-D a^{4} d^{2}-11 D a^{3} b c d +12 D a^{2} b^{2} c^{2}\right ) \sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\left (35 A \,b^{3} d^{2}-15 B a \,b^{2} d^{2}-20 B \,b^{3} c d +3 a^{2} b C \,d^{2}+24 C a \,b^{2} c d +8 C \,b^{3} c^{2}+a^{3} d^{2} D-12 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 d \left (3 A b \,d^{2}-B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-3 D a \,c^{2}\right )}{\left (a d -b c \right )^{4} \sqrt {d x +c}}}{d}\) | \(459\) |
pseudoelliptic | \(\frac {\frac {35 \left (d x +c \right )^{\frac {3}{2}} \left (\left (b^{3} A -\frac {3}{7} a \,b^{2} B +\frac {3}{35} C \,a^{2} b +\frac {1}{35} D a^{3}\right ) d^{2}-\frac {4 \left (B \,b^{2}-\frac {6}{5} C a b +\frac {3}{5} D a^{2}\right ) b c d}{7}+\frac {8 b^{2} c^{2} \left (C b -3 D a \right )}{35}\right ) \left (b x +a \right )^{2} d \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{4}-\frac {2 \left (\left (-\frac {105 A \,x^{3} b^{4}}{8}-\frac {175 a \,x^{2} \left (-\frac {9 B x}{35}+A \right ) b^{3}}{8}-7 a^{2} x \left (\frac {9}{56} C \,x^{2}-\frac {75}{56} B x +A \right ) b^{2}+a^{3} \left (A -\frac {3}{8} D x^{3}-\frac {15}{8} C \,x^{2}+3 B x \right ) b +\frac {3 D a^{4} x^{2}}{8}\right ) d^{4}-10 \left (\frac {7 x^{2} \left (-\frac {3 B x}{7}+A \right ) b^{4}}{4}+\frac {119 a x \left (\frac {36}{119} C \,x^{2}-\frac {80}{119} B x +A \right ) b^{3}}{40}+a^{2} \left (-\frac {9}{20} D x^{3}+\frac {33}{20} C \,x^{2}-\frac {67}{40} B x +A \right ) b^{2}-\frac {a^{3} \left (\frac {3}{2} D x^{2}-\frac {39}{8} C x +B \right ) b}{5}-\frac {3 D a^{4} x}{40}\right ) c \,d^{3}-\frac {39 c^{2} \left (\frac {7 x \left (\frac {8}{7} C \,x^{2}-\frac {80}{21} B x +A \right ) b^{4}}{13}+a \left (-\frac {24}{13} D x^{3}+\frac {136}{39} C \,x^{2}-\frac {145}{39} B x +A \right ) b^{3}-\frac {83 a^{2} \left (\frac {216}{83} D x^{2}-\frac {149}{83} C x +B \right ) b^{2}}{39}+\frac {55 a^{3} \left (-\frac {129 D x}{55}+C \right ) b}{39}-\frac {D a^{4}}{13}\right ) d^{2}}{8}+\frac {3 b \,c^{3} \left (\left (-\frac {16}{3} C \,x^{2}+2 B x +A \right ) b^{3}+a \left (\frac {32}{3} D x^{2}-\frac {44}{3} C x +B \right ) b^{2}-\frac {25 a^{2} \left (-\frac {82 D x}{25}+C \right ) b}{3}+\frac {47 D a^{3}}{3}\right ) d}{4}+D b^{2} c^{4} \left (b x +a \right )^{2}\right ) \sqrt {\left (a d -b c \right ) b}}{3}}{\left (a d -b c \right )^{4} \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{2} \left (d x +c \right )^{\frac {3}{2}} b d}\) | \(502\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1938 vs. \(2 (415) = 830\).
Time = 0.51 (sec) , antiderivative size = 3889, normalized size of antiderivative = 8.88 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.33 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=-\frac {{\left (24 \, D a b^{2} c^{2} - 8 \, C b^{3} c^{2} + 12 \, D a^{2} b c d - 24 \, C a b^{2} c d + 20 \, B b^{3} c d - D a^{3} d^{2} - 3 \, C a^{2} b d^{2} + 15 \, B a b^{2} d^{2} - 35 \, A b^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (D b c^{4} + 9 \, {\left (d x + c\right )} D a c^{2} d - 3 \, {\left (d x + c\right )} C b c^{2} d - D a c^{3} d - C b c^{3} d - 6 \, {\left (d x + c\right )} C a c d^{2} + 6 \, {\left (d x + c\right )} B b c d^{2} + C a c^{2} d^{2} + B b c^{2} d^{2} + 3 \, {\left (d x + c\right )} B a d^{3} - 9 \, {\left (d x + c\right )} A b d^{3} - B a c d^{3} - A b c d^{3} + A a d^{4}\right )}}{3 \, {\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} - \frac {12 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} b^{2} c d - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b^{3} c d + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{4} c d - 12 \, \sqrt {d x + c} D a^{2} b^{2} c^{2} d + 8 \, \sqrt {d x + c} C a b^{3} c^{2} d - 4 \, \sqrt {d x + c} B b^{4} c^{2} d - {\left (d x + c\right )}^{\frac {3}{2}} D a^{3} b d^{2} - 3 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{2} b^{2} d^{2} + 7 \, {\left (d x + c\right )}^{\frac {3}{2}} B a b^{3} d^{2} - 11 \, {\left (d x + c\right )}^{\frac {3}{2}} A b^{4} d^{2} + 11 \, \sqrt {d x + c} D a^{3} b c d^{2} - 3 \, \sqrt {d x + c} C a^{2} b^{2} c d^{2} - 5 \, \sqrt {d x + c} B a b^{3} c d^{2} + 13 \, \sqrt {d x + c} A b^{4} c d^{2} + \sqrt {d x + c} D a^{4} d^{3} - 5 \, \sqrt {d x + c} C a^{3} b d^{3} + 9 \, \sqrt {d x + c} B a^{2} b^{2} d^{3} - 13 \, \sqrt {d x + c} A a b^{3} d^{3}}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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