Optimal. Leaf size=165 \[ -\frac {d^4 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{3/2}}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}-\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c} \]
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Rubi [A] time = 0.10, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac {d^4 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{3/2}}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}-\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 685
Rule 692
Rubi steps
\begin {align*} \int (b d+2 c d x)^4 \sqrt {a+b x+c x^2} \, dx &=\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^4}{\sqrt {a+b x+c x^2}} \, dx}{24 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{32 c}\\ &=-\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (\left (b^2-4 a c\right )^3 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{64 c}\\ &=-\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (\left (b^2-4 a c\right )^3 d^4\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}\\ &=-\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right )^3 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 203, normalized size = 1.23 \[ d^4 \left (\frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \left (2 (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )-\sqrt {c} \sqrt {4 a-\frac {b^2}{c}} \left (4 a c-b^2\right ) \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )\right )}{64 c \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}}+\frac {1}{3} (b+2 c x)^3 (a+x (b+c x))^{3/2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 473, normalized size = 2.87 \[ \left [-\frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} d^{4} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (256 \, c^{6} d^{4} x^{5} + 640 \, b c^{5} d^{4} x^{4} + 16 \, {\left (39 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{3} + 8 \, {\left (37 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{2} + 2 \, {\left (31 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} d^{4} x + {\left (3 \, b^{5} c + 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{2}}, \frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} d^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (256 \, c^{6} d^{4} x^{5} + 640 \, b c^{5} d^{4} x^{4} + 16 \, {\left (39 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{3} + 8 \, {\left (37 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{2} + 2 \, {\left (31 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} d^{4} x + {\left (3 \, b^{5} c + 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{192 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 259, normalized size = 1.57 \[ \frac {1}{96} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, c^{4} d^{4} x + 5 \, b c^{3} d^{4}\right )} x + \frac {39 \, b^{2} c^{7} d^{4} + 4 \, a c^{8} d^{4}}{c^{5}}\right )} x + \frac {37 \, b^{3} c^{6} d^{4} + 12 \, a b c^{7} d^{4}}{c^{5}}\right )} x + \frac {31 \, b^{4} c^{5} d^{4} + 48 \, a b^{2} c^{6} d^{4} - 48 \, a^{2} c^{7} d^{4}}{c^{5}}\right )} x + \frac {3 \, b^{5} c^{4} d^{4} + 32 \, a b^{3} c^{5} d^{4} - 48 \, a^{2} b c^{6} d^{4}}{c^{5}}\right )} + \frac {{\left (b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{64 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 413, normalized size = 2.50 \[ \frac {8 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3} d^{4} x^{3}}{3}+a^{3} c^{\frac {3}{2}} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {3 a^{2} b^{2} \sqrt {c}\, d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4}+\frac {3 a \,b^{4} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}-\frac {b^{6} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {3}{2}}}+\sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2} d^{4} x -\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c \,d^{4} x}{2}+\frac {\sqrt {c \,x^{2}+b x +a}\, b^{4} d^{4} x}{16}+4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,c^{2} d^{4} x^{2}+\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b c \,d^{4}}{2}-\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{3} d^{4}}{4}-2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,c^{2} d^{4} x +\frac {\sqrt {c \,x^{2}+b x +a}\, b^{5} d^{4}}{32 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} c \,d^{4} x}{2}-\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b c \,d^{4}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} d^{4}}{12} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 1144, normalized size = 6.93 \[ \text {result too large to display} \]
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 \[ d^{4} \left (\int b^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 16 c^{4} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 32 b c^{3} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 24 b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 8 b^{3} c x \sqrt {a + b x + c x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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