Optimal. Leaf size=310 \[ -\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{160 a^3 x^4}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}+\frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right )}{512 a^5 x^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{960 a^4 x^3}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6} \]
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Rubi [A] time = 0.36, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {834, 806, 720, 724, 206} \begin {gather*} \frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right )}{512 a^5 x^2}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{960 a^4 x^3}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{160 a^3 x^4}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}-\frac {\int \frac {\left (\frac {3}{2} (3 A b-4 a B)+3 A c x\right ) \sqrt {a+b x+c x^2}}{x^6} \, dx}{6 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}+\frac {\int \frac {\left (\frac {3}{4} \left (21 A b^2-28 a b B-20 a A c\right )+3 (3 A b-4 a B) c x\right ) \sqrt {a+b x+c x^2}}{x^5} \, dx}{30 a^2}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}-\frac {\int \frac {\left (\frac {3}{8} \left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right )+\frac {3}{4} c \left (21 A b^2-28 a b B-20 a A c\right ) x\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx}{120 a^3}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{128 a^4}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}+\frac {\left (\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{1024 a^5}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{512 a^5}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 262, normalized size = 0.85 \begin {gather*} \frac {\frac {2 x^3 (a+x (b+c x))^{3/2} \left (7 A \left (15 b^3-28 a b c\right )+4 a B \left (32 a c-35 b^2\right )\right )}{a^2}+\frac {15 x^4 \left (A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )+4 a b B \left (12 a c-7 b^2\right )\right ) \left (x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}\right )}{8 a^{7/2}}+\frac {12 x^2 (a+x (b+c x))^{3/2} \left (20 a A c+28 a b B-21 A b^2\right )}{a}+96 x (3 A b-4 a B) (a+x (b+c x))^{3/2}-320 a A (a+x (b+c x))^{3/2}}{1920 a^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.05, size = 388, normalized size = 1.25 \begin {gather*} -\frac {21 A b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{512 a^{11/2}}+\frac {\left (-16 a^2 A c^3-48 a^2 b B c^2+60 a A b^2 c^2+40 a b^3 B c-35 A b^4 c-7 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{128 a^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (-1280 a^5 A-1536 a^5 B x-128 a^4 A b x-320 a^4 A c x^2-192 a^4 b B x^2-512 a^4 B c x^3+144 a^3 A b^2 x^2+544 a^3 A b c x^3+480 a^3 A c^2 x^4+224 a^3 b^2 B x^3+928 a^3 b B c x^4+1024 a^3 B c^2 x^5-168 a^2 A b^3 x^3-896 a^2 A b^2 c x^4-1808 a^2 A b c^2 x^5-280 a^2 b^3 B x^4-1840 a^2 b^2 B c x^5+210 a A b^4 x^4+1680 a A b^3 c x^5+420 a b^4 B x^5-315 A b^5 x^5\right )}{7680 a^5 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.40, size = 709, normalized size = 2.29 \begin {gather*} \left [\frac {15 \, {\left (28 \, B a b^{5} - 21 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} c^{2} - 20 \, {\left (8 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} c\right )} \sqrt {a} x^{6} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (1280 \, A a^{6} - {\left (420 \, B a^{2} b^{4} - 315 \, A a b^{5} + 16 \, {\left (64 \, B a^{4} - 113 \, A a^{3} b\right )} c^{2} - 80 \, {\left (23 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} c\right )} x^{5} + 2 \, {\left (140 \, B a^{3} b^{3} - 105 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 16 \, {\left (29 \, B a^{4} b - 28 \, A a^{3} b^{2}\right )} c\right )} x^{4} - 8 \, {\left (28 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3} - 4 \, {\left (16 \, B a^{5} - 17 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (12 \, B a^{5} b - 9 \, A a^{4} b^{2} + 20 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, a^{6} x^{6}}, \frac {15 \, {\left (28 \, B a b^{5} - 21 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} c^{2} - 20 \, {\left (8 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (1280 \, A a^{6} - {\left (420 \, B a^{2} b^{4} - 315 \, A a b^{5} + 16 \, {\left (64 \, B a^{4} - 113 \, A a^{3} b\right )} c^{2} - 80 \, {\left (23 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} c\right )} x^{5} + 2 \, {\left (140 \, B a^{3} b^{3} - 105 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 16 \, {\left (29 \, B a^{4} b - 28 \, A a^{3} b^{2}\right )} c\right )} x^{4} - 8 \, {\left (28 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3} - 4 \, {\left (16 \, B a^{5} - 17 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (12 \, B a^{5} b - 9 \, A a^{4} b^{2} + 20 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, a^{6} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 1955, normalized size = 6.31
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1014, normalized size = 3.27
result too large to display
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 (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^7} \,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 (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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