Optimal. Leaf size=245 \[ \frac {3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{32 c^3}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{4 c} \]
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Rubi [A] time = 0.33, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{32 c^3}+\frac {3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2}}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx &=\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {\int \frac {(d+e x)^2 (3 c (b d-2 a e)+3 c (2 c d-b e) x)}{\sqrt {a+b x+c x^2}} \, dx}{4 c}\\ &=\frac {(2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{4 c}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {\int \frac {(d+e x) \left (\frac {3}{2} c \left (b^2 d e-20 a c d e+4 b \left (c d^2+a e^2\right )\right )+\frac {3}{2} c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{12 c^2}\\ &=\frac {(2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{4 c}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {\left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{32 c^3}+\frac {\left (3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{64 c^3}\\ &=\frac {(2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{4 c}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {\left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{32 c^3}+\frac {\left (3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\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 )}{32 c^3}\\ &=\frac {(2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{4 c}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {\left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{32 c^3}+\frac {3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 302, normalized size = 1.23 \begin {gather*} \frac {-4 a^2 c e^2 (-13 b e+32 c d+6 c e x)+a \left (-15 b^3 e^3+2 b^2 c e^2 (24 d+31 e x)+4 b c^2 e \left (-12 d^2-40 d e x+5 e^2 x^2\right )+8 c^3 \left (8 d^3+12 d^2 e x-8 d e^2 x^2-e^3 x^3\right )\right )+x (b+c x) \left (-15 b^3 e^3+2 b^2 c e^2 (24 d+5 e x)-8 b c^2 e \left (6 d^2+4 d e x+e^2 x^2\right )+16 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )}{32 c^3 \sqrt {a+x (b+c x)}}+\frac {3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{64 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.98, size = 263, normalized size = 1.07 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (52 a b c e^3-128 a c^2 d e^2-24 a c^2 e^3 x-15 b^3 e^3+48 b^2 c d e^2+10 b^2 c e^3 x-48 b c^2 d^2 e-32 b c^2 d e^2 x-8 b c^2 e^3 x^2+64 c^3 d^3+96 c^3 d^2 e x+64 c^3 d e^2 x^2+16 c^3 e^3 x^3\right )}{32 c^3}-\frac {3 \left (16 a^2 c^2 e^3-24 a b^2 c e^3+64 a b c^2 d e^2-64 a c^3 d^2 e+5 b^4 e^3-16 b^3 c d e^2+16 b^2 c^2 d^2 e\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{64 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 545, normalized size = 2.22 \begin {gather*} \left [\frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{3}\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 (16 \, c^{4} e^{3} x^{3} + 64 \, c^{4} d^{3} - 48 \, b c^{3} d^{2} e + 16 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} d e^{2} - {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} e^{3} + 8 \, {\left (8 \, c^{4} d e^{2} - b c^{3} e^{3}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} e - 16 \, b c^{3} d e^{2} + {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{128 \, c^{4}}, -\frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{3}\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 (16 \, c^{4} e^{3} x^{3} + 64 \, c^{4} d^{3} - 48 \, b c^{3} d^{2} e + 16 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} d e^{2} - {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} e^{3} + 8 \, {\left (8 \, c^{4} d e^{2} - b c^{3} e^{3}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} e - 16 \, b c^{3} d e^{2} + {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{64 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 252, normalized size = 1.03 \begin {gather*} \frac {1}{32} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, x e^{3} + \frac {8 \, c^{3} d e^{2} - b c^{2} e^{3}}{c^{3}}\right )} x + \frac {48 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3} - 12 \, a c^{2} e^{3}}{c^{3}}\right )} x + \frac {64 \, c^{3} d^{3} - 48 \, b c^{2} d^{2} e + 48 \, b^{2} c d e^{2} - 128 \, a c^{2} d e^{2} - 15 \, b^{3} e^{3} + 52 \, a b c e^{3}}{c^{3}}\right )} - \frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} e - 64 \, a c^{3} d^{2} e - 16 \, b^{3} c d e^{2} + 64 \, a b c^{2} d e^{2} + 5 \, b^{4} e^{3} - 24 \, a b^{2} c e^{3} + 16 \, a^{2} c^{2} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{64 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 539, normalized size = 2.20 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, e^{3} x^{3}}{2}-\frac {\sqrt {c \,x^{2}+b x +a}\, b \,e^{3} x^{2}}{4 c}+2 \sqrt {c \,x^{2}+b x +a}\, d \,e^{2} x^{2}+\frac {3 a^{2} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {3}{2}}}-\frac {9 a \,b^{2} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}+\frac {3 a b d \,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 {3 a \,d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {15 b^{4} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {7}{2}}}-\frac {3 b^{3} d \,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 {3 b^{2} d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {3}{2}}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,e^{3} x}{4 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{2} e^{3} x}{16 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b d \,e^{2} x}{c}+3 \sqrt {c \,x^{2}+b x +a}\, d^{2} e x +\frac {13 \sqrt {c \,x^{2}+b x +a}\, a b \,e^{3}}{8 c^{2}}-\frac {4 \sqrt {c \,x^{2}+b x +a}\, a d \,e^{2}}{c}-\frac {15 \sqrt {c \,x^{2}+b x +a}\, b^{3} e^{3}}{32 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{2} d \,e^{2}}{2 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b \,d^{2} e}{2 c}+2 \sqrt {c \,x^{2}+b x +a}\, d^{3} \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 (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^3}{\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 (b + 2 c x\right ) \left (d + e x\right )^{3}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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