Integrand size = 22, antiderivative size = 296 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {7 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {e \left (608 c^3 d^3-105 b^3 e^3+20 b c e^2 (24 b d+11 a e)-8 c^2 d e (101 b d+64 a e)+2 c e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (128 c^4 d^4+35 b^4 e^4-128 c^3 d^2 e (2 b d+3 a e)-40 b^2 c e^3 (4 b d+3 a e)+48 c^2 e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}} \]
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Time = 0.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 846, 793, 635, 212} \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right )}{128 c^{9/2}}+\frac {e \sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{192 c^4}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c} \]
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Rule 212
Rule 635
Rule 756
Rule 793
Rule 846
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x)^2 \left (\frac {1}{2} \left (8 c d^2-e (b d+6 a e)\right )+\frac {7}{2} e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{4 c} \\ & = \frac {7 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x) \left (\frac {1}{4} \left (48 c^2 d^3+7 b e^2 (b d+4 a e)-4 c d e (5 b d+23 a e)\right )+\frac {1}{4} e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{12 c^2} \\ & = \frac {7 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {e \left (608 c^3 d^3-105 b^3 e^3+20 b c e^2 (24 b d+11 a e)-8 c^2 d e (101 b d+64 a e)+2 c e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (\frac {3}{8} b^2 e^2 \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right )-\frac {1}{2} a c e^2 \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right )+2 c \left (\frac {1}{2} c d \left (48 c^2 d^3+7 b e^2 (b d+4 a e)-4 c d e (5 b d+23 a e)\right )-b \left (\frac {1}{4} d e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right )+\frac {1}{4} e \left (48 c^2 d^3+7 b e^2 (b d+4 a e)-4 c d e (5 b d+23 a e)\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{48 c^4} \\ & = \frac {7 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {e \left (608 c^3 d^3-105 b^3 e^3+20 b c e^2 (24 b d+11 a e)-8 c^2 d e (101 b d+64 a e)+2 c e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (\frac {3}{8} b^2 e^2 \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right )-\frac {1}{2} a c e^2 \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right )+2 c \left (\frac {1}{2} c d \left (48 c^2 d^3+7 b e^2 (b d+4 a e)-4 c d e (5 b d+23 a e)\right )-b \left (\frac {1}{4} d e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right )+\frac {1}{4} e \left (48 c^2 d^3+7 b e^2 (b d+4 a e)-4 c d e (5 b d+23 a e)\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{24 c^4} \\ & = \frac {7 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {e \left (608 c^3 d^3-105 b^3 e^3+20 b c e^2 (24 b d+11 a e)-8 c^2 d e (101 b d+64 a e)+2 c e \left (104 c^2 d^2+35 b^2 e^2-4 c e (26 b d+9 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (128 c^4 d^4+35 b^4 e^4-128 c^3 d^2 e (2 b d+3 a e)-40 b^2 c e^3 (4 b d+3 a e)+48 c^2 e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} e \sqrt {a+x (b+c x)} \left (-105 b^3 e^3+10 b c e^2 (48 b d+22 a e+7 b e x)+16 c^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )-8 c^2 e \left (a e (64 d+9 e x)+b \left (108 d^2+40 d e x+7 e^2 x^2\right )\right )\right )-3 \left (128 c^4 d^4+35 b^4 e^4-128 c^3 d^2 e (2 b d+3 a e)-40 b^2 c e^3 (4 b d+3 a e)+48 c^2 e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{384 c^{9/2}} \]
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Time = 0.44 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {e \left (48 c^{3} x^{3} e^{3}-56 b \,c^{2} e^{3} x^{2}+256 c^{3} d \,e^{2} x^{2}-72 a \,c^{2} e^{3} x +70 b^{2} c \,e^{3} x -320 b \,c^{2} d \,e^{2} x +576 c^{3} d^{2} e x +220 a b c \,e^{3}-512 a \,c^{2} d \,e^{2}-105 b^{3} e^{3}+480 b^{2} c d \,e^{2}-864 b \,c^{2} d^{2} e +768 c^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{4}}+\frac {\left (48 a^{2} c^{2} e^{4}-120 a \,b^{2} e^{4} c +384 a b \,c^{2} d \,e^{3}-384 a \,c^{3} d^{2} e^{2}+35 b^{4} e^{4}-160 b^{3} c d \,e^{3}+288 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}\) | \(276\) |
default | \(\frac {d^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+e^{4} \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+6 d^{2} e^{2} \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+4 d^{3} e \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )\) | \(717\) |
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Time = 0.36 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.02 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\left [\frac {3 \, {\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 96 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 32 \, {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d e^{3} + {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} e^{4} x^{3} + 768 \, c^{4} d^{3} e - 864 \, b c^{3} d^{2} e^{2} + 32 \, {\left (15 \, b^{2} c^{2} - 16 \, a c^{3}\right )} d e^{3} - 5 \, {\left (21 \, b^{3} c - 44 \, a b c^{2}\right )} e^{4} + 8 \, {\left (32 \, c^{4} d e^{3} - 7 \, b c^{3} e^{4}\right )} x^{2} + 2 \, {\left (288 \, c^{4} d^{2} e^{2} - 160 \, b c^{3} d e^{3} + {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{5}}, -\frac {3 \, {\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 96 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 32 \, {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d e^{3} + {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (48 \, c^{4} e^{4} x^{3} + 768 \, c^{4} d^{3} e - 864 \, b c^{3} d^{2} e^{2} + 32 \, {\left (15 \, b^{2} c^{2} - 16 \, a c^{3}\right )} d e^{3} - 5 \, {\left (21 \, b^{3} c - 44 \, a b c^{2}\right )} e^{4} + 8 \, {\left (32 \, c^{4} d e^{3} - 7 \, b c^{3} e^{4}\right )} x^{2} + 2 \, {\left (288 \, c^{4} d^{2} e^{2} - 160 \, b c^{3} d e^{3} + {\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (299) = 598\).
Time = 0.80 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {e^{4} x^{3}}{4 c} + \frac {x^{2} \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{3 c} + \frac {x \left (- \frac {3 a e^{4}}{4 c} - \frac {5 b \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{6 c} + 6 d^{2} e^{2}\right )}{2 c} + \frac {- \frac {2 a \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{3 c} - \frac {3 b \left (- \frac {3 a e^{4}}{4 c} - \frac {5 b \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{6 c} + 6 d^{2} e^{2}\right )}{4 c} + 4 d^{3} e}{c}\right ) + \left (- \frac {a \left (- \frac {3 a e^{4}}{4 c} - \frac {5 b \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{6 c} + 6 d^{2} e^{2}\right )}{2 c} - \frac {b \left (- \frac {2 a \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{3 c} - \frac {3 b \left (- \frac {3 a e^{4}}{4 c} - \frac {5 b \left (- \frac {7 b e^{4}}{8 c} + 4 d e^{3}\right )}{6 c} + 6 d^{2} e^{2}\right )}{4 c} + 4 d^{3} e\right )}{2 c} + d^{4}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {e^{4} \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 a e^{4} + 4 b d e^{3}\right )}{7 b^{4}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (6 a^{2} e^{4} - 12 a b d e^{3} + 6 b^{2} d^{2} e^{2}\right )}{5 b^{4}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 4 a^{3} e^{4} + 12 a^{2} b d e^{3} - 12 a b^{2} d^{2} e^{2} + 4 b^{3} d^{3} e\right )}{3 b^{4}} + \frac {\sqrt {a + b x} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{b^{4}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {\begin {cases} d^{4} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{5}}{5 e} & \text {otherwise} \end {cases}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (\frac {6 \, e^{4} x}{c} + \frac {32 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}}{c^{4}}\right )} x + \frac {288 \, c^{3} d^{2} e^{2} - 160 \, b c^{2} d e^{3} + 35 \, b^{2} c e^{4} - 36 \, a c^{2} e^{4}}{c^{4}}\right )} x + \frac {768 \, c^{3} d^{3} e - 864 \, b c^{2} d^{2} e^{2} + 480 \, b^{2} c d e^{3} - 512 \, a c^{2} d e^{3} - 105 \, b^{3} e^{4} + 220 \, a b c e^{4}}{c^{4}}\right )} - \frac {{\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 288 \, b^{2} c^{2} d^{2} e^{2} - 384 \, a c^{3} d^{2} e^{2} - 160 \, b^{3} c d e^{3} + 384 \, a b c^{2} d e^{3} + 35 \, b^{4} e^{4} - 120 \, a b^{2} c e^{4} + 48 \, a^{2} c^{2} e^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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