Optimal. Leaf size=212 \[ \frac {\left (b^2-4 a c\right )^2 \left (-4 a c C+24 A c^2+7 b^2 C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{192 c^3}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.18, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1661, 640, 612, 621, 206} \[ \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{192 c^3}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{512 c^4}+\frac {\left (b^2-4 a c\right )^2 \left (-4 a c C+24 A c^2+7 b^2 C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx &=\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (6 A c-a C-\frac {7 b C x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\frac {7 b^2 C}{2}+2 c (6 A c-a C)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac {\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\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 )}{512 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 267, normalized size = 1.26 \[ \frac {\frac {360 A \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{c^{3/2}}+1920 A (b+2 c x) (a+x (b+c x))^{3/2}+\frac {C \left (5 \left (7 b^2-4 a c\right ) \left (\frac {3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{c^{5/2}}+\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )-1792 b (a+x (b+c x))^{5/2}\right )}{c}+2560 C x (a+x (b+c x))^{5/2}}{15360 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 605, normalized size = 2.85 \[ \left [\frac {15 \, {\left (7 \, C b^{6} - 60 \, C a b^{4} c + 384 \, A a^{2} c^{4} - 64 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c^{3} + 24 \, {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (1280 \, C c^{6} x^{5} + 1664 \, C b c^{5} x^{4} - 105 \, C b^{5} c + 760 \, C a b^{3} c^{2} + 2400 \, A a b c^{4} - 72 \, {\left (18 \, C a^{2} b + 5 \, A b^{3}\right )} c^{3} + 16 \, {\left (3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}\right )} x^{3} - 8 \, {\left (7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}\right )} x^{2} + 2 \, {\left (35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 2400 \, A a c^{5} + 120 \, {\left (2 \, C a^{2} + A b^{2}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (7 \, C b^{6} - 60 \, C a b^{4} c + 384 \, A a^{2} c^{4} - 64 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c^{3} + 24 \, {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (1280 \, C c^{6} x^{5} + 1664 \, C b c^{5} x^{4} - 105 \, C b^{5} c + 760 \, C a b^{3} c^{2} + 2400 \, A a b c^{4} - 72 \, {\left (18 \, C a^{2} b + 5 \, A b^{3}\right )} c^{3} + 16 \, {\left (3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}\right )} x^{3} - 8 \, {\left (7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}\right )} x^{2} + 2 \, {\left (35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 2400 \, A a c^{5} + 120 \, {\left (2 \, C a^{2} + A b^{2}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 297, normalized size = 1.40 \[ \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, C c x + 13 \, C b\right )} x + \frac {3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}}{c^{5}}\right )} x - \frac {7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}}{c^{5}}\right )} x + \frac {35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 240 \, C a^{2} c^{4} + 120 \, A b^{2} c^{4} + 2400 \, A a c^{5}}{c^{5}}\right )} x - \frac {105 \, C b^{5} c - 760 \, C a b^{3} c^{2} + 1296 \, C a^{2} b c^{3} + 360 \, A b^{3} c^{3} - 2400 \, A a b c^{4}}{c^{5}}\right )} - \frac {{\left (7 \, C b^{6} - 60 \, C a b^{4} c + 144 \, C a^{2} b^{2} c^{2} + 24 \, A b^{4} c^{2} - 64 \, C a^{3} c^{3} - 192 \, A a b^{2} c^{3} + 384 \, A a^{2} c^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 613, normalized size = 2.89 \[ \frac {3 A \,a^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}-\frac {3 A a \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 A \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {5}{2}}}-\frac {C \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {9 C \,a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}-\frac {15 C a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {7 C \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a x}{8}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} x}{32 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, C \,a^{2} x}{16 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, C a \,b^{2} x}{8 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, C \,b^{4} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a b}{16 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A x}{4}-\frac {\sqrt {c \,x^{2}+b x +a}\, C \,a^{2} b}{32 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, C a \,b^{3}}{16 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} C a x}{24 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, C \,b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} C \,b^{2} x}{96 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{8 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} C a b}{48 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} C \,b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} C x}{6 c}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} C b}{60 c^{2}} \]
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 \[ \int \left (C\,x^2+A\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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 \[ \int \left (A + C x^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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