Optimal. Leaf size=296 \[ \frac {\left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}}+\frac {e \sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{192 c^4}+\frac {7 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{24 c^2}+\frac {e (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c} \]
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Rubi [A] time = 0.38, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {742, 832, 779, 621, 206} \begin {gather*} \frac {\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 ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 742
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\sqrt {a+b x+c x^2}} \, dx &=\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 ) \operatorname {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*}
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Mathematica [A] time = 0.46, size = 352, normalized size = 1.19 \begin {gather*} \frac {\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 ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{128 c^{9/2}}+\frac {e \left (-4 a^2 c e^2 (2 c (64 d+9 e x)-55 b e)+a \left (-105 b^3 e^3+10 b^2 c e^2 (48 d+29 e x)+4 b c^2 e \left (-216 d^2-208 d e x+23 e^2 x^2\right )+8 c^3 \left (96 d^3+72 d^2 e x-32 d e^2 x^2-3 e^3 x^3\right )\right )+x (b+c x) \left (-105 b^3 e^3+10 b^2 c e^2 (48 d+7 e x)-8 b c^2 e \left (108 d^2+40 d e x+7 e^2 x^2\right )+16 c^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )\right )}{192 c^4 \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.93, size = 290, normalized size = 0.98 \begin {gather*} \frac {\left (-48 a^2 c^2 e^4+120 a b^2 c e^4-384 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 ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{128 c^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (220 a b c e^4-512 a c^2 d e^3-72 a c^2 e^4 x-105 b^3 e^4+480 b^2 c d e^3+70 b^2 c e^4 x-864 b c^2 d^2 e^2-320 b c^2 d e^3 x-56 b c^2 e^4 x^2+768 c^3 d^3 e+576 c^3 d^2 e^2 x+256 c^3 d e^3 x^2+48 c^3 e^4 x^3\right )}{192 c^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 599, normalized size = 2.02 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 276, normalized size = 0.93 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, x {\left (\frac {6 \, x e^{4}}{c} + \frac {32 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}}{c^{4}}\right )} + \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 627, normalized size = 2.12 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, e^{4} x^{3}}{4 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, b \,e^{4} x^{2}}{24 c^{2}}+\frac {4 \sqrt {c \,x^{2}+b x +a}\, d \,e^{3} x^{2}}{3 c}+\frac {3 a^{2} e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {15 a \,b^{2} e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {3 a b d \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {5}{2}}}-\frac {3 a \,d^{2} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}+\frac {35 b^{4} e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {5 b^{3} d \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {7}{2}}}+\frac {9 b^{2} d^{2} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {2 b \,d^{3} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}+\frac {d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,e^{4} x}{8 c^{2}}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, b^{2} e^{4} x}{96 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, b d \,e^{3} x}{3 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, d^{2} e^{2} x}{c}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, a b \,e^{4}}{48 c^{3}}-\frac {8 \sqrt {c \,x^{2}+b x +a}\, a d \,e^{3}}{3 c^{2}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, b^{3} e^{4}}{64 c^{4}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{2} d \,e^{3}}{2 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, b \,d^{2} e^{2}}{2 c^{2}}+\frac {4 \sqrt {c \,x^{2}+b x +a}\, d^{3} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{4}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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