Optimal. Leaf size=267 \[ -\frac {2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac {2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac {2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac {2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac {2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac {2 B c^2 (d+e x)^{15/2}}{15 e^6} \]
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Rubi [A] time = 0.16, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac {2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac {2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac {2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac {2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac {2 B c^2 (d+e x)^{15/2}}{15 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{3/2}}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{5/2}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{7/2}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{9/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{11/2}}{e^5}+\frac {B c^2 (d+e x)^{13/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{5/2}}{5 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{7/2}}{7 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{13/2}}{13 e^6}+\frac {2 B c^2 (d+e x)^{15/2}}{15 e^6}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 272, normalized size = 1.02 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (A e \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+B \left (39 b^2 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+6 b c e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+c^2 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 399, normalized size = 1.49 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 A b^2 d^2 e^3-12870 A b^2 d e^3 (d+e x)+5005 A b^2 e^3 (d+e x)^2-18018 A b c d^3 e^2+38610 A b c d^2 e^2 (d+e x)-30030 A b c d e^2 (d+e x)^2+8190 A b c e^2 (d+e x)^3+9009 A c^2 d^4 e-25740 A c^2 d^3 e (d+e x)+30030 A c^2 d^2 e (d+e x)^2-16380 A c^2 d e (d+e x)^3+3465 A c^2 e (d+e x)^4-9009 b^2 B d^3 e^2+19305 b^2 B d^2 e^2 (d+e x)-15015 b^2 B d e^2 (d+e x)^2+4095 b^2 B e^2 (d+e x)^3+18018 b B c d^4 e-51480 b B c d^3 e (d+e x)+60060 b B c d^2 e (d+e x)^2-32760 b B c d e (d+e x)^3+6930 b B c e (d+e x)^4-9009 B c^2 d^5+32175 B c^2 d^4 (d+e x)-50050 B c^2 d^3 (d+e x)^2+40950 B c^2 d^2 (d+e x)^3-17325 B c^2 d (d+e x)^4+3003 B c^2 (d+e x)^5\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 425, normalized size = 1.59 \begin {gather*} \frac {2 \, {\left (3003 \, B c^{2} e^{7} x^{7} - 256 \, B c^{2} d^{7} + 1144 \, A b^{2} d^{4} e^{3} + 384 \, {\left (2 \, B b c + A c^{2}\right )} d^{6} e - 624 \, {\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{2} + 231 \, {\left (16 \, B c^{2} d e^{6} + 15 \, {\left (2 \, B b c + A c^{2}\right )} e^{7}\right )} x^{6} + 63 \, {\left (B c^{2} d^{2} e^{5} + 70 \, {\left (2 \, B b c + A c^{2}\right )} d e^{6} + 65 \, {\left (B b^{2} + 2 \, A b c\right )} e^{7}\right )} x^{5} - 35 \, {\left (2 \, B c^{2} d^{3} e^{4} - 143 \, A b^{2} e^{7} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{5} - 156 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{6}\right )} x^{4} + 5 \, {\left (16 \, B c^{2} d^{4} e^{3} + 1430 \, A b^{2} d e^{6} - 24 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{4} + 39 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{5}\right )} x^{3} - 3 \, {\left (32 \, B c^{2} d^{5} e^{2} - 143 \, A b^{2} d^{2} e^{5} - 48 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{3} + 78 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{4}\right )} x^{2} + 4 \, {\left (32 \, B c^{2} d^{6} e - 143 \, A b^{2} d^{3} e^{4} - 48 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e^{2} + 78 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 1382, normalized size = 5.18
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 341, normalized size = 1.28 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 B \,c^{2} x^{5} e^{5}+3465 A \,c^{2} e^{5} x^{4}+6930 B b c \,e^{5} x^{4}-2310 B \,c^{2} d \,e^{4} x^{4}+8190 A b c \,e^{5} x^{3}-2520 A \,c^{2} d \,e^{4} x^{3}+4095 B \,b^{2} e^{5} x^{3}-5040 B b c d \,e^{4} x^{3}+1680 B \,c^{2} d^{2} e^{3} x^{3}+5005 A \,b^{2} e^{5} x^{2}-5460 A b c d \,e^{4} x^{2}+1680 A \,c^{2} d^{2} e^{3} x^{2}-2730 B \,b^{2} d \,e^{4} x^{2}+3360 B b c \,d^{2} e^{3} x^{2}-1120 B \,c^{2} d^{3} e^{2} x^{2}-2860 A \,b^{2} d \,e^{4} x +3120 A b c \,d^{2} e^{3} x -960 A \,c^{2} d^{3} e^{2} x +1560 B \,b^{2} d^{2} e^{3} x -1920 B b c \,d^{3} e^{2} x +640 B \,c^{2} d^{4} e x +1144 A \,b^{2} d^{2} e^{3}-1248 A b c \,d^{3} e^{2}+384 A \,c^{2} d^{4} e -624 B \,b^{2} d^{3} e^{2}+768 B b c \,d^{4} e -256 B \,c^{2} d^{5}\right )}{45045 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 291, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B c^{2} - 3465 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 4095 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 9009 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 254, normalized size = 0.95 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{9\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{11\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{7\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 33.46, size = 937, normalized size = 3.51
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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