Optimal. Leaf size=203 \[ \frac {5 \left (b^2-4 a c\right )^3 (b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}-\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c} \]
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Rubi [A] time = 0.10, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \begin {gather*} \frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac {5 \left (b^2-4 a c\right )^3 (b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rubi steps
\begin {align*} \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {(-b B+2 A c) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{2 c}\\ &=-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 \left (b^2-4 a c\right ) (b B-2 A c)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=\frac {5 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right )^2 (b B-2 A c)\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}+\frac {5 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 \left (b^2-4 a c\right )^3 (b B-2 A c)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^4}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}+\frac {5 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 \left (b^2-4 a c\right )^3 (b B-2 A c)\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^4}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}+\frac {5 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {5 \left (b^2-4 a c\right )^3 (b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 179, normalized size = 0.88 \begin {gather*} \frac {B (a+x (b+c x))^{7/2}}{7 c}-\frac {(b B-2 A c) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\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 )\right )\right )\right )}{6144 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.72, size = 422, normalized size = 2.08 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (3072 a^3 B c^3+7392 a^2 A b c^3+14784 a^2 A c^4 x-3696 a^2 b^2 B c^2+1824 a^2 b B c^3 x+9216 a^2 B c^4 x^2-2240 a A b^3 c^2+1344 a A b^2 c^3 x+17472 a A b c^4 x^2+11648 a A c^5 x^3+1120 a b^4 B c-672 a b^3 B c^2 x+480 a b^2 B c^3 x^2+12608 a b B c^4 x^3+9216 a B c^5 x^4+210 A b^5 c-140 A b^4 c^2 x+112 A b^3 c^3 x^2+6048 A b^2 c^4 x^3+8960 A b c^5 x^4+3584 A c^6 x^5-105 b^6 B+70 b^5 B c x-56 b^4 B c^2 x^2+48 b^3 B c^3 x^3+4736 b^2 B c^4 x^4+7424 b B c^5 x^5+3072 B c^6 x^6\right )}{21504 c^4}-\frac {5 \left (128 a^3 A c^4-64 a^3 b B c^3-96 a^2 A b^2 c^3+48 a^2 b^3 B c^2+24 a A b^4 c^2-12 a b^5 B c-2 A b^6 c+b^7 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 843, normalized size = 4.15 \begin {gather*} \left [-\frac {105 \, {\left (B b^{7} + 128 \, A a^{3} c^{4} - 32 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 24 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\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 (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 96 \, {\left (32 \, B a^{3} + 77 \, A a^{2} b\right )} c^{4} + 128 \, {\left (37 \, B b^{2} c^{5} + 2 \, {\left (36 \, B a + 35 \, A b\right )} c^{6}\right )} x^{4} - 112 \, {\left (33 \, B a^{2} b^{2} + 20 \, A a b^{3}\right )} c^{3} + 16 \, {\left (3 \, B b^{3} c^{4} + 728 \, A a c^{6} + 2 \, {\left (394 \, B a b + 189 \, A b^{2}\right )} c^{5}\right )} x^{3} + 70 \, {\left (16 \, B a b^{4} + 3 \, A b^{5}\right )} c^{2} - 8 \, {\left (7 \, B b^{4} c^{3} - 24 \, {\left (48 \, B a^{2} + 91 \, A a b\right )} c^{5} - 2 \, {\left (30 \, B a b^{2} + 7 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{5} c^{2} + 7392 \, A a^{2} c^{5} + 48 \, {\left (19 \, B a^{2} b + 14 \, A a b^{2}\right )} c^{4} - 14 \, {\left (24 \, B a b^{3} + 5 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{86016 \, c^{5}}, -\frac {105 \, {\left (B b^{7} + 128 \, A a^{3} c^{4} - 32 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 24 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\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 (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 96 \, {\left (32 \, B a^{3} + 77 \, A a^{2} b\right )} c^{4} + 128 \, {\left (37 \, B b^{2} c^{5} + 2 \, {\left (36 \, B a + 35 \, A b\right )} c^{6}\right )} x^{4} - 112 \, {\left (33 \, B a^{2} b^{2} + 20 \, A a b^{3}\right )} c^{3} + 16 \, {\left (3 \, B b^{3} c^{4} + 728 \, A a c^{6} + 2 \, {\left (394 \, B a b + 189 \, A b^{2}\right )} c^{5}\right )} x^{3} + 70 \, {\left (16 \, B a b^{4} + 3 \, A b^{5}\right )} c^{2} - 8 \, {\left (7 \, B b^{4} c^{3} - 24 \, {\left (48 \, B a^{2} + 91 \, A a b\right )} c^{5} - 2 \, {\left (30 \, B a b^{2} + 7 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{5} c^{2} + 7392 \, A a^{2} c^{5} + 48 \, {\left (19 \, B a^{2} b + 14 \, A a b^{2}\right )} c^{4} - 14 \, {\left (24 \, B a b^{3} + 5 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{43008 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 425, normalized size = 2.09 \begin {gather*} \frac {1}{21504} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, B c^{2} x + \frac {29 \, B b c^{7} + 14 \, A c^{8}}{c^{6}}\right )} x + \frac {37 \, B b^{2} c^{6} + 72 \, B a c^{7} + 70 \, A b c^{7}}{c^{6}}\right )} x + \frac {3 \, B b^{3} c^{5} + 788 \, B a b c^{6} + 378 \, A b^{2} c^{6} + 728 \, A a c^{7}}{c^{6}}\right )} x - \frac {7 \, B b^{4} c^{4} - 60 \, B a b^{2} c^{5} - 14 \, A b^{3} c^{5} - 1152 \, B a^{2} c^{6} - 2184 \, A a b c^{6}}{c^{6}}\right )} x + \frac {35 \, B b^{5} c^{3} - 336 \, B a b^{3} c^{4} - 70 \, A b^{4} c^{4} + 912 \, B a^{2} b c^{5} + 672 \, A a b^{2} c^{5} + 7392 \, A a^{2} c^{6}}{c^{6}}\right )} x - \frac {105 \, B b^{6} c^{2} - 1120 \, B a b^{4} c^{3} - 210 \, A b^{5} c^{3} + 3696 \, B a^{2} b^{2} c^{4} + 2240 \, A a b^{3} c^{4} - 3072 \, B a^{3} c^{5} - 7392 \, A a^{2} b c^{5}}{c^{6}}\right )} - \frac {5 \, {\left (B b^{7} - 12 \, B a b^{5} c - 2 \, A b^{6} c + 48 \, B a^{2} b^{3} c^{2} + 24 \, A a b^{4} c^{2} - 64 \, B a^{3} b c^{3} - 96 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{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 {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 807, normalized size = 3.98 \begin {gather*} \frac {5 A \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}-\frac {15 A \,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 {3}{2}}}+\frac {15 A a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {5}{2}}}-\frac {5 A \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {7}{2}}}-\frac {5 B \,a^{3} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {3}{2}}}+\frac {15 B \,a^{2} b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {5}{2}}}-\frac {15 B a \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {7}{2}}}+\frac {5 B \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {9}{2}}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,a^{2} x}{16}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{2} x}{32 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4} x}{256 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b x}{32 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3} x}{64 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5} x}{512 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,a^{2} b}{32 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{3}}{64 c^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a x}{24}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{5}}{512 c^{3}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} x}{96 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b^{2}}{64 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{4}}{128 c^{3}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b x}{48 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{6}}{1024 c^{4}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3} x}{192 c^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a b}{48 c}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{192 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A x}{6}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,b^{2}}{96 c^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{4}}{384 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b x}{12 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{12 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{24 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} B}{7 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 \left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,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 \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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