Integrand size = 32, antiderivative size = 375 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {\left (b^3 (6 B c-5 A d)-a b^2 (12 c C+B d)-13 a^3 d D+a^2 b (7 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^3 (b c-a d)^2 (a+b x)^2}-\frac {\left (b^3 \left (8 c^2 C-6 B c d+5 A d^2\right )-11 a^3 d^2 D+a^2 b d (C d+30 c D)-a b^2 \left (4 c C d-B d^2+24 c^2 D\right )\right ) \sqrt {c+d x}}{8 b^3 (b c-a d)^3 (a+b x)}+\frac {\left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (4 c C d-B d^2-24 c^2 D\right )+b^3 \left (8 c^2 C d-6 B c d^2+5 A d^3-16 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{7/2} (b c-a d)^{7/2}} \]
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Time = 0.56 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1635, 911, 1171, 393, 214} \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}+\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (-B d^2-24 c^2 D+4 c C d\right )+b^3 \left (5 A d^3-6 B c d^2-16 c^3 D+8 c^2 C d\right )\right )}{8 b^{7/2} (b c-a d)^{7/2}}-\frac {\sqrt {c+d x} \left (-11 a^3 d^2 D+a^2 b d (30 c D+C d)-a b^2 \left (-B d^2+24 c^2 D+4 c C d\right )+b^3 \left (5 A d^2-6 B c d+8 c^2 C\right )\right )}{8 b^3 (a+b x) (b c-a d)^3}-\frac {\sqrt {c+d x} \left (-13 a^3 d D+a^2 b (18 c D+7 C d)-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right )}{12 b^3 (a+b x)^2 (b c-a d)^2} \]
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Rule 214
Rule 393
Rule 911
Rule 1171
Rule 1635
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {\int \frac {-\frac {b^3 (6 B c-5 A d)-a b^2 (6 c C+B d)-a^3 d D+a^2 b (C d+6 c D)}{2 b^3}-\frac {3 (b c-a d) (b C-a D) x}{b^2}-3 \left (c-\frac {a d}{b}\right ) D x^2}{(a+b x)^3 \sqrt {c+d x}} \, dx}{3 (b c-a d)} \\ & = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {2 \text {Subst}\left (\int \frac {\frac {-3 c^2 \left (c-\frac {a d}{b}\right ) D+\frac {3 c d (b c-a d) (b C-a D)}{b^2}-\frac {d^2 \left (b^3 (6 B c-5 A d)-a b^2 (6 c C+B d)-a^3 d D+a^2 b (C d+6 c D)\right )}{2 b^3}}{d^2}-\frac {\left (-6 c \left (c-\frac {a d}{b}\right ) D+\frac {3 d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac {3 \left (c-\frac {a d}{b}\right ) D x^4}{d^2}}{\left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )^3} \, dx,x,\sqrt {c+d x}\right )}{3 d (b c-a d)} \\ & = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {\left (b^3 (6 B c-5 A d)-a b^2 (12 c C+B d)-13 a^3 d D+a^2 b (7 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^3 (b c-a d)^2 (a+b x)^2}-\frac {\text {Subst}\left (\int \frac {\frac {3 \left (a b^2 d^2 (4 c C-B d)+3 a^3 d^3 D-a^2 b d^2 (C d+6 c D)-b^3 \left (8 c^2 C d-6 B c d^2+5 A d^3-8 c^3 D\right )\right )}{2 b^3 d^2}-\frac {12 (b c-a d)^2 D x^2}{b^2 d^2}}{\left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )^2} \, dx,x,\sqrt {c+d x}\right )}{6 (b c-a d)^2} \\ & = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {\left (b^3 (6 B c-5 A d)-a b^2 (12 c C+B d)-13 a^3 d D+a^2 b (7 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^3 (b c-a d)^2 (a+b x)^2}-\frac {\left (b^3 \left (8 c^2 C-6 B c d+5 A d^2\right )-11 a^3 d^2 D+a^2 b d (C d+30 c D)-a b^2 \left (4 c C d-B d^2+24 c^2 D\right )\right ) \sqrt {c+d x}}{8 b^3 (b c-a d)^3 (a+b x)}-\frac {\left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (4 c C d-B d^2-24 c^2 D\right )+b^3 \left (8 c^2 C d-6 B c d^2+5 A d^3-16 c^3 D\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {-b c+a d}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 b^3 d (b c-a d)^3} \\ & = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {\left (b^3 (6 B c-5 A d)-a b^2 (12 c C+B d)-13 a^3 d D+a^2 b (7 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^3 (b c-a d)^2 (a+b x)^2}-\frac {\left (b^3 \left (8 c^2 C-6 B c d+5 A d^2\right )-11 a^3 d^2 D+a^2 b d (C d+30 c D)-a b^2 \left (4 c C d-B d^2+24 c^2 D\right )\right ) \sqrt {c+d x}}{8 b^3 (b c-a d)^3 (a+b x)}+\frac {\left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (4 c C d-B d^2-24 c^2 D\right )+b^3 \left (8 c^2 C d-6 B c d^2+5 A d^3-16 c^3 D\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{7/2} (b c-a d)^{7/2}} \\ \end{align*}
Time = 1.23 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x} \left (-15 a^5 d^2 D+6 b^5 c x (2 B c+4 c C x-3 B d x)+a^4 b d (-3 C d+44 c D-40 d D x)+a b^4 \left (-12 c x (-2 c C+C d x+6 c D x)+B \left (4 c^2-50 c d x+3 d^2 x^2\right )\right )+A b^3 \left (33 a^2 d^2+2 a b d (-13 c+20 d x)+b^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )-a^3 b^2 \left (44 c^2 D-2 c d (5 C+59 D x)+d^2 \left (3 B+8 C x+33 D x^2\right )\right )+a^2 b^3 \left (d^2 x (8 B+3 C x)+4 c^2 (2 C-27 D x)+2 c d \left (-8 B+7 C x+45 D x^2\right )\right )\right )}{24 b^3 (b c-a d)^3 (a+b x)^3}-\frac {\left (-5 a^3 d^3 D+a^2 b d^2 (-C d+18 c D)-a b^2 d \left (-4 c C d+B d^2+24 c^2 D\right )+b^3 \left (-8 c^2 C d+6 B c d^2-5 A d^3+16 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{7/2} (-b c+a d)^{7/2}} \]
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Time = 1.83 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {\frac {5 \left (\left (A \,d^{3}-\frac {6}{5} B c \,d^{2}+\frac {8}{5} C \,c^{2} d -\frac {16}{5} D c^{3}\right ) b^{3}+\frac {a d \left (B \,d^{2}-4 C c d +24 D c^{2}\right ) b^{2}}{5}+\frac {a^{2} b \,d^{2} \left (C d -18 D c \right )}{5}+a^{3} d^{3} D\right ) \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8}+\frac {11 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}\, \left (\left (\frac {5 A \,d^{2} x^{2}}{11}-\frac {10 x c \left (\frac {9 B x}{5}+A \right ) d}{33}+\frac {8 \left (3 C \,x^{2}+\frac {3}{2} B x +A \right ) c^{2}}{33}\right ) b^{5}-\frac {26 a \left (-\frac {20 \left (\frac {3 B x}{40}+A \right ) x \,d^{2}}{13}+c \left (\frac {6}{13} C \,x^{2}+\frac {25}{13} B x +A \right ) d -\frac {2 c^{2} \left (-18 D x^{2}+6 C x +B \right )}{13}\right ) b^{4}}{33}+a^{2} \left (\left (A +\frac {8}{33} B x +\frac {1}{11} C \,x^{2}\right ) d^{2}-\frac {16 \left (-\frac {45}{8} D x^{2}-\frac {7}{8} C x +B \right ) c d}{33}+\frac {8 \left (-\frac {27 D x}{2}+C \right ) c^{2}}{33}\right ) b^{3}-\frac {\left (\left (11 D x^{2}+\frac {8}{3} C x +B \right ) d^{2}-\frac {10 c \left (\frac {59 D x}{5}+C \right ) d}{3}+\frac {44 D c^{2}}{3}\right ) a^{3} b^{2}}{11}-\frac {a^{4} \left (\left (\frac {40 D x}{3}+C \right ) d -\frac {44 D c}{3}\right ) d b}{11}-\frac {5 D a^{5} d^{2}}{11}\right )}{8}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{3} \left (a d -b c \right )^{3} b^{3}}\) | \(382\) |
derivativedivides | \(\frac {\frac {d \left (5 A \,b^{3} d^{2}+B a \,b^{2} d^{2}-6 B \,b^{3} c d +a^{2} b C \,d^{2}-4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-11 a^{3} d^{2} D+30 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{8 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (5 A \,b^{3} d^{2}+B a \,b^{2} d^{2}-6 B \,b^{3} c d -a^{2} b C \,d^{2}+6 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 D a^{2} b c d -18 D a \,b^{2} c^{2}\right ) d \left (d x +c \right )^{\frac {3}{2}}}{3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (11 A \,b^{3} d^{2}-B a \,b^{2} d^{2}-10 B \,b^{3} c d -a^{2} b C \,d^{2}+4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) d \sqrt {d x +c}}{8 b^{3} \left (a d -b c \right )}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\left (5 A \,b^{3} d^{3}+B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}+a^{2} b C \,d^{3}-4 C a \,b^{2} c \,d^{2}+8 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-18 D a^{2} b c \,d^{2}+24 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\) | \(548\) |
default | \(\frac {\frac {d \left (5 A \,b^{3} d^{2}+B a \,b^{2} d^{2}-6 B \,b^{3} c d +a^{2} b C \,d^{2}-4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-11 a^{3} d^{2} D+30 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{8 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (5 A \,b^{3} d^{2}+B a \,b^{2} d^{2}-6 B \,b^{3} c d -a^{2} b C \,d^{2}+6 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 D a^{2} b c d -18 D a \,b^{2} c^{2}\right ) d \left (d x +c \right )^{\frac {3}{2}}}{3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (11 A \,b^{3} d^{2}-B a \,b^{2} d^{2}-10 B \,b^{3} c d -a^{2} b C \,d^{2}+4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) d \sqrt {d x +c}}{8 b^{3} \left (a d -b c \right )}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\left (5 A \,b^{3} d^{3}+B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}+a^{2} b C \,d^{3}-4 C a \,b^{2} c \,d^{2}+8 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-18 D a^{2} b c \,d^{2}+24 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\) | \(548\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (354) = 708\).
Time = 0.37 (sec) , antiderivative size = 2446, normalized size of antiderivative = 6.52 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 976 vs. \(2 (354) = 708\).
Time = 0.31 (sec) , antiderivative size = 976, normalized size of antiderivative = 2.60 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {{\left (16 \, D b^{3} c^{3} - 24 \, D a b^{2} c^{2} d - 8 \, C b^{3} c^{2} d + 18 \, D a^{2} b c d^{2} + 4 \, C a b^{2} c d^{2} + 6 \, B b^{3} c d^{2} - 5 \, D a^{3} d^{3} - C a^{2} b d^{3} - B a b^{2} d^{3} - 5 \, A b^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {72 \, {\left (d x + c\right )}^{\frac {5}{2}} D a b^{4} c^{2} d - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} C b^{5} c^{2} d - 144 \, {\left (d x + c\right )}^{\frac {3}{2}} D a b^{4} c^{3} d + 48 \, {\left (d x + c\right )}^{\frac {3}{2}} C b^{5} c^{3} d + 72 \, \sqrt {d x + c} D a b^{4} c^{4} d - 24 \, \sqrt {d x + c} C b^{5} c^{4} d - 90 \, {\left (d x + c\right )}^{\frac {5}{2}} D a^{2} b^{3} c d^{2} + 12 \, {\left (d x + c\right )}^{\frac {5}{2}} C a b^{4} c d^{2} + 18 \, {\left (d x + c\right )}^{\frac {5}{2}} B b^{5} c d^{2} + 288 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} b^{3} c^{2} d^{2} - 48 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b^{4} c^{2} d^{2} - 48 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{5} c^{2} d^{2} - 198 \, \sqrt {d x + c} D a^{2} b^{3} c^{3} d^{2} + 36 \, \sqrt {d x + c} C a b^{4} c^{3} d^{2} + 30 \, \sqrt {d x + c} B b^{5} c^{3} d^{2} + 33 \, {\left (d x + c\right )}^{\frac {5}{2}} D a^{3} b^{2} d^{3} - 3 \, {\left (d x + c\right )}^{\frac {5}{2}} C a^{2} b^{3} d^{3} - 3 \, {\left (d x + c\right )}^{\frac {5}{2}} B a b^{4} d^{3} - 15 \, {\left (d x + c\right )}^{\frac {5}{2}} A b^{5} d^{3} - 184 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{3} b^{2} c d^{3} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{2} b^{3} c d^{3} + 56 \, {\left (d x + c\right )}^{\frac {3}{2}} B a b^{4} c d^{3} + 40 \, {\left (d x + c\right )}^{\frac {3}{2}} A b^{5} c d^{3} + 195 \, \sqrt {d x + c} D a^{3} b^{2} c^{2} d^{3} + 3 \, \sqrt {d x + c} C a^{2} b^{3} c^{2} d^{3} - 57 \, \sqrt {d x + c} B a b^{4} c^{2} d^{3} - 33 \, \sqrt {d x + c} A b^{5} c^{2} d^{3} + 40 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{4} b d^{4} + 8 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{3} b^{2} d^{4} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{4} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} A a b^{4} d^{4} - 84 \, \sqrt {d x + c} D a^{4} b c d^{4} - 18 \, \sqrt {d x + c} C a^{3} b^{2} c d^{4} + 24 \, \sqrt {d x + c} B a^{2} b^{3} c d^{4} + 66 \, \sqrt {d x + c} A a b^{4} c d^{4} + 15 \, \sqrt {d x + c} D a^{5} d^{5} + 3 \, \sqrt {d x + c} C a^{4} b d^{5} + 3 \, \sqrt {d x + c} B a^{3} b^{2} d^{5} - 33 \, \sqrt {d x + c} A a^{2} b^{3} d^{5}}{24 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^4\,\sqrt {c+d\,x}} \,d x \]
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