Optimal. Leaf size=252 \[ -\frac {5 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{6144 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{384 c^3}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{384 c^3}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{6144 c^4}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rubi steps
\begin {align*} \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{32 c^2}\\ &=\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B 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 )}{16384 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 205, normalized size = 0.81 \begin {gather*} \frac {\frac {7 \left (-2 a B c-8 A b c+\frac {9 b^2 B}{2}\right ) \left (2 (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (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 )}{128 c^{5/2}}+\frac {(b+2 c x) (a+x (b+c x))^{3/2}}{8 c}\right )\right )}{24 c}+(a+x (b+c x))^{7/2} (2 c (8 A+7 B x)-9 b B)}{112 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.45, size = 535, normalized size = 2.12 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (49152 a^3 A c^4-42432 a^3 b B c^3+13440 a^3 B c^4 x-59136 a^2 A b^2 c^3+29184 a^2 A b c^4 x+147456 a^2 A c^5 x^2+37744 a^2 b^3 B c^2-19104 a^2 b^2 B c^3 x+11136 a^2 b B c^4 x^2+105728 a^2 B c^5 x^3+17920 a A b^4 c^2-10752 a A b^3 c^3 x+7680 a A b^2 c^4 x^2+201728 a A b c^5 x^3+147456 a A c^6 x^4-10500 a b^5 B c+6328 a b^4 B c^2 x-4544 a b^3 B c^3 x^2+3456 a b^2 B c^4 x^3+157184 a b B c^5 x^4+121856 a B c^6 x^5-1680 A b^6 c+1120 A b^5 c^2 x-896 A b^4 c^3 x^2+768 A b^3 c^4 x^3+75776 A b^2 c^5 x^4+118784 A b c^6 x^5+49152 A c^7 x^6+945 b^7 B-630 b^6 B c x+504 b^5 B c^2 x^2-432 b^4 B c^3 x^3+384 b^3 B c^4 x^4+62208 b^2 B c^5 x^5+101376 b B c^6 x^6+43008 B c^7 x^7\right )}{344064 c^5}+\frac {5 \left (256 a^4 B c^4+1024 a^3 A b c^4-768 a^3 b^2 B c^3-768 a^2 A b^3 c^3+480 a^2 b^4 B c^2+192 a A b^5 c^2-112 a b^6 B c-16 A b^7 c+9 b^8 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{32768 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 1039, normalized size = 4.12
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 528, normalized size = 2.10 \begin {gather*} \frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, B c^{2} x + \frac {33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac {243 \, B b^{2} c^{7} + 476 \, B a c^{8} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac {3 \, B b^{3} c^{6} + 1228 \, B a b c^{7} + 592 \, A b^{2} c^{7} + 1152 \, A a c^{8}}{c^{7}}\right )} x - \frac {27 \, B b^{4} c^{5} - 216 \, B a b^{2} c^{6} - 48 \, A b^{3} c^{6} - 6608 \, B a^{2} c^{7} - 12608 \, A a b c^{7}}{c^{7}}\right )} x + \frac {63 \, B b^{5} c^{4} - 568 \, B a b^{3} c^{5} - 112 \, A b^{4} c^{5} + 1392 \, B a^{2} b c^{6} + 960 \, A a b^{2} c^{6} + 18432 \, A a^{2} c^{7}}{c^{7}}\right )} x - \frac {315 \, B b^{6} c^{3} - 3164 \, B a b^{4} c^{4} - 560 \, A b^{5} c^{4} + 9552 \, B a^{2} b^{2} c^{5} + 5376 \, A a b^{3} c^{5} - 6720 \, B a^{3} c^{6} - 14592 \, A a^{2} b c^{6}}{c^{7}}\right )} x + \frac {945 \, B b^{7} c^{2} - 10500 \, B a b^{5} c^{3} - 1680 \, A b^{6} c^{3} + 37744 \, B a^{2} b^{3} c^{4} + 17920 \, A a b^{4} c^{4} - 42432 \, B a^{3} b c^{5} - 59136 \, A a^{2} b^{2} c^{5} + 49152 \, A a^{3} c^{6}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, B b^{8} - 112 \, B a b^{6} c - 16 \, A b^{7} c + 480 \, B a^{2} b^{4} c^{2} + 192 \, A a b^{5} c^{2} - 768 \, B a^{3} b^{2} c^{3} - 768 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1034, normalized size = 4.10
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 x\,\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 x \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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