Integrand size = 32, antiderivative size = 188 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=\frac {2 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) \sqrt {c+d x}}{b^3 d^3}+\frac {2 (b C d-2 b c D-a d D) (c+d x)^{3/2}}{3 b^2 d^3}+\frac {2 D (c+d x)^{5/2}}{5 b d^3}-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2} \sqrt {b c-a d}} \]
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Time = 0.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1634, 65, 214} \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=-\frac {2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2} \sqrt {b c-a d}}+\frac {2 \sqrt {c+d x} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3}+\frac {2 (c+d x)^{3/2} (-a d D-2 b c D+b C d)}{3 b^2 d^3}+\frac {2 D (c+d x)^{5/2}}{5 b d^3} \]
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Rule 65
Rule 214
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )}{b^3 d^2 \sqrt {c+d x}}+\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{b^3 (a+b x) \sqrt {c+d x}}+\frac {(b C d-2 b c D-a d D) \sqrt {c+d x}}{b^2 d^2}+\frac {D (c+d x)^{3/2}}{b d^2}\right ) \, dx \\ & = \frac {2 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) \sqrt {c+d x}}{b^3 d^3}+\frac {2 (b C d-2 b c D-a d D) (c+d x)^{3/2}}{3 b^2 d^3}+\frac {2 D (c+d x)^{5/2}}{5 b d^3}+\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx \\ & = \frac {2 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) \sqrt {c+d x}}{b^3 d^3}+\frac {2 (b C d-2 b c D-a d D) (c+d x)^{3/2}}{3 b^2 d^3}+\frac {2 D (c+d x)^{5/2}}{5 b d^3}+\frac {\left (2 \left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right )\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = \frac {2 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) \sqrt {c+d x}}{b^3 d^3}+\frac {2 (b C d-2 b c D-a d D) (c+d x)^{3/2}}{3 b^2 d^3}+\frac {2 D (c+d x)^{5/2}}{5 b d^3}-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (15 a^2 d^2 D-5 a b d (3 C d-2 c D+d D x)+b^2 \left (8 c^2 D-2 c d (5 C+2 D x)+d^2 \left (15 B+5 C x+3 D x^2\right )\right )\right )}{15 b^3 d^3}+\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{7/2} \sqrt {-b c+a d}} \]
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Time = 1.72 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {2 d^{3} \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+2 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}\, \left (\left (\left (\frac {1}{5} D x^{2}+\frac {1}{3} C x +B \right ) b^{2}-a \left (\frac {D x}{3}+C \right ) b +D a^{2}\right ) d^{2}-\frac {2 \left (\left (\frac {2 D x}{5}+C \right ) b -D a \right ) b c d}{3}+\frac {8 D b^{2} c^{2}}{15}\right )}{d^{3} b^{3} \sqrt {\left (a d -b c \right ) b}}\) | \(157\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {D \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}+\frac {C \,b^{2} d \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {D a b d \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {2 D b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+B \,b^{2} d^{2} \sqrt {d x +c}-C a b \,d^{2} \sqrt {d x +c}-C \,b^{2} c d \sqrt {d x +c}+D a^{2} d^{2} \sqrt {d x +c}+D a b c d \sqrt {d x +c}+D b^{2} c^{2} \sqrt {d x +c}\right )}{b^{3}}+\frac {2 d^{3} \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{3} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(220\) |
default | \(\frac {\frac {2 \left (\frac {D \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}+\frac {C \,b^{2} d \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {D a b d \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {2 D b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+B \,b^{2} d^{2} \sqrt {d x +c}-C a b \,d^{2} \sqrt {d x +c}-C \,b^{2} c d \sqrt {d x +c}+D a^{2} d^{2} \sqrt {d x +c}+D a b c d \sqrt {d x +c}+D b^{2} c^{2} \sqrt {d x +c}\right )}{b^{3}}+\frac {2 d^{3} \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{3} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(220\) |
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Time = 0.27 (sec) , antiderivative size = 565, normalized size of antiderivative = 3.01 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=\left [\frac {15 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \sqrt {b^{2} c - a b d} d^{3} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (8 \, D b^{4} c^{3} - 15 \, {\left (D a^{3} b - C a^{2} b^{2} + B a b^{3}\right )} d^{3} + 5 \, {\left (D a^{2} b^{2} c - {\left (C a b^{3} - 3 \, B b^{4}\right )} c\right )} d^{2} + 3 \, {\left (D b^{4} c d^{2} - D a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (D a b^{3} c^{2} - 5 \, C b^{4} c^{2}\right )} d - {\left (4 \, D b^{4} c^{2} d - 5 \, {\left (D a^{2} b^{2} - C a b^{3}\right )} d^{3} + {\left (D a b^{3} c - 5 \, C b^{4} c\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )}}, -\frac {2 \, {\left (15 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \sqrt {-b^{2} c + a b d} d^{3} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (8 \, D b^{4} c^{3} - 15 \, {\left (D a^{3} b - C a^{2} b^{2} + B a b^{3}\right )} d^{3} + 5 \, {\left (D a^{2} b^{2} c - {\left (C a b^{3} - 3 \, B b^{4}\right )} c\right )} d^{2} + 3 \, {\left (D b^{4} c d^{2} - D a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (D a b^{3} c^{2} - 5 \, C b^{4} c^{2}\right )} d - {\left (4 \, D b^{4} c^{2} d - 5 \, {\left (D a^{2} b^{2} - C a b^{3}\right )} d^{3} + {\left (D a b^{3} c - 5 \, C b^{4} c\right )} d^{2}\right )} x\right )} \sqrt {d x + c}\right )}}{15 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )}}\right ] \]
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Time = 3.94 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.46 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\frac {D \left (c + d x\right )^{\frac {5}{2}}}{5 b d^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (C b d - D a d - 2 D b c\right )}{3 b^{2} d^{2}} + \frac {\sqrt {c + d x} \left (B b^{2} d^{2} - C a b d^{2} - C b^{2} c d + D a^{2} d^{2} + D a b c d + D b^{2} c^{2}\right )}{b^{3} d^{2}} - \frac {d \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{4} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\frac {D x^{3}}{3 b} + \frac {x^{2} \left (C b - D a\right )}{2 b^{2}} + \frac {x \left (B b^{2} - C a b + D a^{2}\right )}{b^{3}} - \frac {\left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{3}}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} D b^{4} d^{12} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} D b^{4} c d^{12} + 15 \, \sqrt {d x + c} D b^{4} c^{2} d^{12} - 5 \, {\left (d x + c\right )}^{\frac {3}{2}} D a b^{3} d^{13} + 5 \, {\left (d x + c\right )}^{\frac {3}{2}} C b^{4} d^{13} + 15 \, \sqrt {d x + c} D a b^{3} c d^{13} - 15 \, \sqrt {d x + c} C b^{4} c d^{13} + 15 \, \sqrt {d x + c} D a^{2} b^{2} d^{14} - 15 \, \sqrt {d x + c} C a b^{3} d^{14} + 15 \, \sqrt {d x + c} B b^{4} d^{14}\right )}}{15 \, b^{5} d^{15}} \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) \sqrt {c+d x}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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