Optimal. Leaf size=207 \[ \frac {3 d^4 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{5/2}}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}+\frac {3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c} \]
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Rubi [A] time = 0.13, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}+\frac {3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^2}+\frac {3 d^4 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{5/2}}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c} \]
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 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^4 \sqrt {a+b x+c x^2} \, dx}{32 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^4}{\sqrt {a+b x+c x^2}} \, dx}{256 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac {\left (3 \left (b^2-4 a c\right )^3 d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^2}\\ &=\frac {3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^2}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac {\left (3 \left (b^2-4 a c\right )^4 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^2}\\ &=\frac {3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^2}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac {\left (3 \left (b^2-4 a c\right )^4 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 )}{1024 c^2}\\ &=\frac {3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^2}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac {3 \left (b^2-4 a c\right )^4 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 2.00, size = 249, normalized size = 1.20 \[ \frac {1}{4} d^4 \left ((b+2 c x)^3 (a+x (b+c x))^{5/2}-2 c \left (a-\frac {b^2}{4 c}\right ) (b+2 c x) \sqrt {a+x (b+c x)} \left ((a+x (b+c x))^2-\frac {(a+x (b+c x)) \left (2 (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \sqrt {c} \sqrt {4 a-\frac {b^2}{c}} \left (4 a c-b^2\right ) \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )\right )}{256 c (b+2 c x) \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 677, normalized size = 3.27 \[ \left [\frac {3 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c} d^{4} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2048 \, c^{8} d^{4} x^{7} + 7168 \, b c^{7} d^{4} x^{6} + 768 \, {\left (13 \, b^{2} c^{6} + 4 \, a c^{7}\right )} d^{4} x^{5} + 640 \, {\left (11 \, b^{3} c^{5} + 12 \, a b c^{6}\right )} d^{4} x^{4} + 16 \, {\left (161 \, b^{4} c^{4} + 472 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 24 \, {\left (17 \, b^{5} c^{3} + 152 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 2 \, {\left (b^{6} c^{2} + 396 \, a b^{4} c^{3} + 240 \, a^{2} b^{2} c^{4} - 192 \, a^{3} c^{5}\right )} d^{4} x - {\left (3 \, b^{7} c - 44 \, a b^{5} c^{2} - 176 \, a^{2} b^{3} c^{3} + 192 \, a^{3} b c^{4}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{4096 \, c^{3}}, -\frac {3 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-c} d^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (2048 \, c^{8} d^{4} x^{7} + 7168 \, b c^{7} d^{4} x^{6} + 768 \, {\left (13 \, b^{2} c^{6} + 4 \, a c^{7}\right )} d^{4} x^{5} + 640 \, {\left (11 \, b^{3} c^{5} + 12 \, a b c^{6}\right )} d^{4} x^{4} + 16 \, {\left (161 \, b^{4} c^{4} + 472 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 24 \, {\left (17 \, b^{5} c^{3} + 152 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 2 \, {\left (b^{6} c^{2} + 396 \, a b^{4} c^{3} + 240 \, a^{2} b^{2} c^{4} - 192 \, a^{3} c^{5}\right )} d^{4} x - {\left (3 \, b^{7} c - 44 \, a b^{5} c^{2} - 176 \, a^{2} b^{3} c^{3} + 192 \, a^{3} b c^{4}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{2048 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 391, normalized size = 1.89 \[ \frac {1}{1024} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (2 \, c^{5} d^{4} x + 7 \, b c^{4} d^{4}\right )} x + \frac {3 \, {\left (13 \, b^{2} c^{10} d^{4} + 4 \, a c^{11} d^{4}\right )}}{c^{7}}\right )} x + \frac {5 \, {\left (11 \, b^{3} c^{9} d^{4} + 12 \, a b c^{10} d^{4}\right )}}{c^{7}}\right )} x + \frac {161 \, b^{4} c^{8} d^{4} + 472 \, a b^{2} c^{9} d^{4} + 16 \, a^{2} c^{10} d^{4}}{c^{7}}\right )} x + \frac {3 \, {\left (17 \, b^{5} c^{7} d^{4} + 152 \, a b^{3} c^{8} d^{4} + 16 \, a^{2} b c^{9} d^{4}\right )}}{c^{7}}\right )} x + \frac {b^{6} c^{6} d^{4} + 396 \, a b^{4} c^{7} d^{4} + 240 \, a^{2} b^{2} c^{8} d^{4} - 192 \, a^{3} c^{9} d^{4}}{c^{7}}\right )} x - \frac {3 \, b^{7} c^{5} d^{4} - 44 \, a b^{5} c^{6} d^{4} - 176 \, a^{2} b^{3} c^{7} d^{4} + 192 \, a^{3} b c^{8} d^{4}}{c^{7}}\right )} - \frac {3 \, {\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 641, normalized size = 3.10 \[ \frac {3 a^{4} c^{\frac {3}{2}} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8}-\frac {3 a^{3} b^{2} \sqrt {c}\, d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8}+\frac {9 a^{2} b^{4} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 \sqrt {c}}-\frac {3 a \,b^{6} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {3}{2}}}+\frac {3 b^{8} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {5}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} d^{4} x}{8}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,d^{4} x}{32}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} d^{4} x}{128}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{6} d^{4} x}{512 c}+2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{3} d^{4} x^{3}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a^{3} b c \,d^{4}}{16}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} d^{4}}{64}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} c^{2} d^{4} x}{4}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} d^{4}}{256 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} c \,d^{4} x}{8}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{7} d^{4}}{1024 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} d^{4} x}{64}+3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,c^{2} d^{4} x^{2}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b c \,d^{4}}{8}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{3} d^{4}}{16}-\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,c^{2} d^{4} x +\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{5} d^{4}}{128 c}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} c \,d^{4} x}{4}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b c \,d^{4}}{2}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{3} d^{4}}{8} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (b\,d+2\,c\,d\,x\right )}^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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 \[ d^{4} \left (\int a b^{4} \sqrt {a + b x + c x^{2}}\, dx + \int b^{5} x \sqrt {a + b x + c x^{2}}\, dx + \int 16 c^{5} x^{6} \sqrt {a + b x + c x^{2}}\, dx + \int 16 a c^{4} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 48 b c^{4} x^{5} \sqrt {a + b x + c x^{2}}\, dx + \int 56 b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 32 b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 9 b^{4} c x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 32 a b c^{3} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 24 a b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 8 a b^{3} c x \sqrt {a + b x + c x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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