Optimal. Leaf size=345 \[ \frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}-\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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Rubi [A] time = 0.34, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {832, 779, 612, 620, 206} \begin {gather*} -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 779
Rule 832
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\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (-\frac {1}{2} (5 b B-14 A c) d+\frac {1}{2} (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^4}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 1.60, size = 415, normalized size = 1.20 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (14 A c \left (\frac {5}{4} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (3 b^{9/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+b c x \sqrt {\frac {c x}{b}+1} \left (-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right )\right )+320 b c^4 e x^3 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (d+e x)-224 b c^3 e x^3 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (b e-2 c d)\right )+B \left (\frac {7}{4} \left (9 b^2 e^2-28 b c d e+24 c^2 d^2\right ) \left (b c x \sqrt {\frac {c x}{b}+1} \left (15 b^4-10 b^3 c x+8 b^2 c^2 x^2+176 b c^3 x^3+128 c^4 x^4\right )-15 b^{11/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )+3840 b c^5 e x^4 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (d+e x)+320 b c^4 e x^4 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (16 c d-9 b e)\right )\right )}{26880 b c^6 x \sqrt {\frac {c x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.26, size = 537, normalized size = 1.56 \begin {gather*} \frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-14 A b^6 c e^2+48 A b^5 c^2 d e-48 A b^4 c^3 d^2+9 b^7 B e^2-28 b^6 B c d e+24 b^5 B c^2 d^2\right )}{2048 c^{11/2}}+\frac {\sqrt {b x+c x^2} \left (-1470 A b^5 c e^2+5040 A b^4 c^2 d e+980 A b^4 c^2 e^2 x-5040 A b^3 c^3 d^2-3360 A b^3 c^3 d e x-784 A b^3 c^3 e^2 x^2+3360 A b^2 c^4 d^2 x+2688 A b^2 c^4 d e x^2+672 A b^2 c^4 e^2 x^3+40320 A b c^5 d^2 x^2+59136 A b c^5 d e x^3+23296 A b c^5 e^2 x^4+26880 A c^6 d^2 x^3+43008 A c^6 d e x^4+17920 A c^6 e^2 x^5+945 b^6 B e^2-2940 b^5 B c d e-630 b^5 B c e^2 x+2520 b^4 B c^2 d^2+1960 b^4 B c^2 d e x+504 b^4 B c^2 e^2 x^2-1680 b^3 B c^3 d^2 x-1568 b^3 B c^3 d e x^2-432 b^3 B c^3 e^2 x^3+1344 b^2 B c^4 d^2 x^2+1344 b^2 B c^4 d e x^3+384 b^2 B c^4 e^2 x^4+29568 b B c^5 d^2 x^3+46592 b B c^5 d e x^4+19200 b B c^5 e^2 x^5+21504 B c^6 d^2 x^4+35840 B c^6 d e x^5+15360 B c^6 e^2 x^6\right )}{107520 c^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 987, normalized size = 2.86
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 518, normalized size = 1.50 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x e^{2} + \frac {28 \, B c^{7} d e + 15 \, B b c^{6} e^{2} + 14 \, A c^{7} e^{2}}{c^{6}}\right )} x + \frac {168 \, B c^{7} d^{2} + 364 \, B b c^{6} d e + 336 \, A c^{7} d e + 3 \, B b^{2} c^{5} e^{2} + 182 \, A b c^{6} e^{2}}{c^{6}}\right )} x + \frac {3 \, {\left (616 \, B b c^{6} d^{2} + 560 \, A c^{7} d^{2} + 28 \, B b^{2} c^{5} d e + 1232 \, A b c^{6} d e - 9 \, B b^{3} c^{4} e^{2} + 14 \, A b^{2} c^{5} e^{2}\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (24 \, B b^{2} c^{5} d^{2} + 720 \, A b c^{6} d^{2} - 28 \, B b^{3} c^{4} d e + 48 \, A b^{2} c^{5} d e + 9 \, B b^{4} c^{3} e^{2} - 14 \, A b^{3} c^{4} e^{2}\right )}}{c^{6}}\right )} x - \frac {35 \, {\left (24 \, B b^{3} c^{4} d^{2} - 48 \, A b^{2} c^{5} d^{2} - 28 \, B b^{4} c^{3} d e + 48 \, A b^{3} c^{4} d e + 9 \, B b^{5} c^{2} e^{2} - 14 \, A b^{4} c^{3} e^{2}\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (24 \, B b^{4} c^{3} d^{2} - 48 \, A b^{3} c^{4} d^{2} - 28 \, B b^{5} c^{2} d e + 48 \, A b^{4} c^{3} d e + 9 \, B b^{6} c e^{2} - 14 \, A b^{5} c^{2} e^{2}\right )}}{c^{6}}\right )} + \frac {{\left (24 \, B b^{5} c^{2} d^{2} - 48 \, A b^{4} c^{3} d^{2} - 28 \, B b^{6} c d e + 48 \, A b^{5} c^{2} d e + 9 \, B b^{7} e^{2} - 14 \, A b^{6} c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2048 \, 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 = 949, normalized size = 2.75 \begin {gather*} \frac {7 A \,b^{6} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {3 A \,b^{5} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {7}{2}}}+\frac {3 A \,b^{4} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {5}{2}}}-\frac {9 B \,b^{7} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}+\frac {7 B \,b^{6} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{512 c^{\frac {9}{2}}}-\frac {3 B \,b^{5} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A \,b^{4} e^{2} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} d e x}{32 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{2} d^{2} x}{32 c}+\frac {9 \sqrt {c \,x^{2}+b x}\, B \,b^{5} e^{2} x}{512 c^{4}}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{4} d e x}{128 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,b^{3} d^{2} x}{64 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A \,b^{5} e^{2}}{512 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{4} d e}{64 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} d^{2}}{64 c^{2}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} e^{2} x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b d e x}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,d^{2} x}{4}+\frac {9 \sqrt {c \,x^{2}+b x}\, B \,b^{6} e^{2}}{1024 c^{5}}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{5} d e}{256 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,b^{4} d^{2}}{128 c^{3}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} e^{2} x}{64 c^{3}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} d e x}{48 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b \,d^{2} x}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,e^{2} x^{2}}{7 c}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3} e^{2}}{192 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} d e}{8 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b \,d^{2}}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,e^{2} x}{6 c}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4} e^{2}}{128 c^{4}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} d e}{96 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} d^{2}}{16 c^{2}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b \,e^{2} x}{28 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B d e x}{3 c}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b \,e^{2}}{60 c^{2}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A d e}{5 c}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2} e^{2}}{40 c^{3}}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b d e}{30 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,d^{2}}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 728, normalized size = 2.11 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B e^{2} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2} d^{2} x}{32 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{5} e^{2} x}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e^{2} x}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b e^{2} x}{28 \, c^{2}} + \frac {3 \, A b^{4} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {9 \, B b^{7} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b d^{2}}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{6} e^{2}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} e^{2}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} e^{2}}{40 \, c^{3}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{4} x}{256 \, c^{3}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} x}{96 \, c^{2}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{2}} + \frac {{\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}} x}{6 \, c} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{8 \, c} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{5}}{512 \, c^{4}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}}{192 \, c^{3}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{3}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b}{60 \, c^{2}} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{16 \, c^{2}} + \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{5 \, c} \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 (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^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 (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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