Optimal. Leaf size=404 \[ \frac {\left (b x+c x^2\right )^{3/2} \left (6 c e x \left (28 A c e (2 c d-b e)+B \left (21 b^2 e^2-36 b c d e+8 c^2 d^2\right )\right )+4 A c e \left (35 b^2 e^2-150 b c d e+192 c^2 d^2\right )+B \left (-105 b^3 e^3+420 b^2 c d e^2-456 b c^2 d^2 e+64 c^3 d^3\right )\right )}{960 c^4}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-28 b^3 c e^2 (A e+3 B d)+120 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^{11/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} \left (-28 b^3 c e^2 (A e+3 B d)+120 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^5}+\frac {\left (b x+c x^2\right )^{3/2} (d+e x)^2 (4 A c e-3 b B e+2 B c d)}{20 c^2}+\frac {B \left (b x+c x^2\right )^{3/2} (d+e x)^3}{6 c} \]
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Rubi [A] time = 0.55, antiderivative size = 404, 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 {\left (b x+c x^2\right )^{3/2} \left (6 c e x \left (28 A c e (2 c d-b e)+B \left (21 b^2 e^2-36 b c d e+8 c^2 d^2\right )\right )+4 A c e \left (35 b^2 e^2-150 b c d e+192 c^2 d^2\right )+B \left (420 b^2 c d e^2-105 b^3 e^3-456 b c^2 d^2 e+64 c^3 d^3\right )\right )}{960 c^4}+\frac {(b+2 c x) \sqrt {b x+c x^2} \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^5}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^{11/2}}+\frac {\left (b x+c x^2\right )^{3/2} (d+e x)^2 (4 A c e-3 b B e+2 B c d)}{20 c^2}+\frac {B \left (b x+c x^2\right )^{3/2} (d+e x)^3}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^3 \sqrt {b x+c x^2} \, dx &=\frac {B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac {\int (d+e x)^2 \left (-\frac {3}{2} (b B-4 A c) d+\frac {3}{2} (2 B c d-3 b B e+4 A c e) x\right ) \sqrt {b x+c x^2} \, dx}{6 c}\\ &=\frac {(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac {\int (d+e x) \left (-\frac {3}{4} d \left (16 b B c d-40 A c^2 d-9 b^2 B e+12 A b c e\right )+\frac {3}{4} \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2} \, dx}{30 c^2}\\ &=\frac {(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac {\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \int \sqrt {b x+c x^2} \, dx}{128 c^4}\\ &=\frac {\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^5}+\frac {(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac {\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (b^2 \left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^5}\\ &=\frac {\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^5}+\frac {(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac {\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (b^2 \left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 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 )}{512 c^5}\\ &=\frac {\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^5}+\frac {(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac {\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}-\frac {b^2 \left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 422, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-210 b^4 c e^2 (2 A e+6 B d+B e x)+8 b^3 c^2 e \left (5 A e (45 d+7 e x)+3 B \left (75 d^2+35 d e x+7 e^2 x^2\right )\right )-16 b^2 c^3 \left (A e \left (180 d^2+75 d e x+14 e^2 x^2\right )+B \left (60 d^3+75 d^2 e x+42 d e^2 x^2+9 e^3 x^3\right )\right )+64 b c^4 \left (3 A \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+15 d^2 e x+9 d e^2 x^2+2 e^3 x^3\right )\right )+128 c^5 x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+315 b^5 B e^3\right )-\frac {15 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (-28 b^3 c e^2 (A e+3 B d)+120 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{7680 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.14, size = 559, normalized size = 1.38 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-420 A b^4 c e^3+1800 A b^3 c^2 d e^2+280 A b^3 c^2 e^3 x-2880 A b^2 c^3 d^2 e-1200 A b^2 c^3 d e^2 x-224 A b^2 c^3 e^3 x^2+1920 A b c^4 d^3+1920 A b c^4 d^2 e x+960 A b c^4 d e^2 x^2+192 A b c^4 e^3 x^3+3840 A c^5 d^3 x+7680 A c^5 d^2 e x^2+5760 A c^5 d e^2 x^3+1536 A c^5 e^3 x^4+315 b^5 B e^3-1260 b^4 B c d e^2-210 b^4 B c e^3 x+1800 b^3 B c^2 d^2 e+840 b^3 B c^2 d e^2 x+168 b^3 B c^2 e^3 x^2-960 b^2 B c^3 d^3-1200 b^2 B c^3 d^2 e x-672 b^2 B c^3 d e^2 x^2-144 b^2 B c^3 e^3 x^3+640 b B c^4 d^3 x+960 b B c^4 d^2 e x^2+576 b B c^4 d e^2 x^3+128 b B c^4 e^3 x^4+2560 B c^5 d^3 x^2+5760 B c^5 d^2 e x^3+4608 B c^5 d e^2 x^4+1280 B c^5 e^3 x^5\right )}{7680 c^5}+\frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-28 A b^5 c e^3+120 A b^4 c^2 d e^2-192 A b^3 c^3 d^2 e+128 A b^2 c^4 d^3+21 b^6 B e^3-84 b^5 B c d e^2+120 b^4 B c^2 d^2 e-64 b^3 B c^3 d^3\right )}{1024 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 1022, normalized size = 2.53
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 521, normalized size = 1.29 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B x e^{3} + \frac {36 \, B c^{5} d e^{2} + B b c^{4} e^{3} + 12 \, A c^{5} e^{3}}{c^{5}}\right )} x + \frac {3 \, {\left (120 \, B c^{5} d^{2} e + 12 \, B b c^{4} d e^{2} + 120 \, A c^{5} d e^{2} - 3 \, B b^{2} c^{3} e^{3} + 4 \, A b c^{4} e^{3}\right )}}{c^{5}}\right )} x + \frac {320 \, B c^{5} d^{3} + 120 \, B b c^{4} d^{2} e + 960 \, A c^{5} d^{2} e - 84 \, B b^{2} c^{3} d e^{2} + 120 \, A b c^{4} d e^{2} + 21 \, B b^{3} c^{2} e^{3} - 28 \, A b^{2} c^{3} e^{3}}{c^{5}}\right )} x + \frac {5 \, {\left (64 \, B b c^{4} d^{3} + 384 \, A c^{5} d^{3} - 120 \, B b^{2} c^{3} d^{2} e + 192 \, A b c^{4} d^{2} e + 84 \, B b^{3} c^{2} d e^{2} - 120 \, A b^{2} c^{3} d e^{2} - 21 \, B b^{4} c e^{3} + 28 \, A b^{3} c^{2} e^{3}\right )}}{c^{5}}\right )} x - \frac {15 \, {\left (64 \, B b^{2} c^{3} d^{3} - 128 \, A b c^{4} d^{3} - 120 \, B b^{3} c^{2} d^{2} e + 192 \, A b^{2} c^{3} d^{2} e + 84 \, B b^{4} c d e^{2} - 120 \, A b^{3} c^{2} d e^{2} - 21 \, B b^{5} e^{3} + 28 \, A b^{4} c e^{3}\right )}}{c^{5}}\right )} - \frac {{\left (64 \, B b^{3} c^{3} d^{3} - 128 \, A b^{2} c^{4} d^{3} - 120 \, B b^{4} c^{2} d^{2} e + 192 \, A b^{3} c^{3} d^{2} e + 84 \, B b^{5} c d e^{2} - 120 \, A b^{4} c^{2} d e^{2} - 21 \, B b^{6} e^{3} + 28 \, A b^{5} c e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{1024 \, 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 = 1027, normalized size = 2.54
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 760, normalized size = 1.88 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B e^{3} x^{3}}{6 \, c} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b e^{3} x^{2}}{20 \, c^{2}} + \frac {1}{2} \, \sqrt {c x^{2} + b x} A d^{3} x + \frac {21 \, \sqrt {c x^{2} + b x} B b^{4} e^{3} x}{256 \, c^{4}} + \frac {21 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} e^{3} x}{160 \, c^{3}} - \frac {A b^{2} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} - \frac {21 \, B b^{6} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {11}{2}}} + \frac {\sqrt {c x^{2} + b x} A b d^{3}}{4 \, c} + \frac {21 \, \sqrt {c x^{2} + b x} B b^{5} e^{3}}{512 \, c^{5}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e^{3}}{64 \, c^{4}} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{2}}{5 \, c} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{3}} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{40 \, c^{2}} + \frac {15 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b^{2} x}{32 \, c^{2}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} x}{4 \, c} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x} b x}{4 \, c} + \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {9}{2}}} - \frac {15 \, {\left (B d^{2} e + A d e^{2}\right )} b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{4}} + \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{48 \, c^{3}} + \frac {15 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b^{3}}{64 \, c^{3}} - \frac {5 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b}{8 \, c^{2}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{2}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.84, size = 827, normalized size = 2.05 \begin {gather*} A\,d^3\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {7\,A\,b\,e^3\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}+\frac {A\,e^3\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}+\frac {B\,e^3\,x^3\,{\left (c\,x^2+b\,x\right )}^{3/2}}{6\,c}-\frac {A\,b^2\,d^3\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {B\,b^3\,d^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {B\,d^3\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}+\frac {3\,B\,b\,e^3\,\left (\frac {7\,b\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}-\frac {x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}\right )}{4\,c}+\frac {3\,A\,b^3\,d^2\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {A\,d^2\,e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{8\,c^2}+\frac {3\,A\,d\,e^2\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}+\frac {3\,B\,d^2\,e\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {15\,A\,b\,d\,e^2\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}-\frac {15\,B\,b\,d^2\,e\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}-\frac {21\,B\,b\,d\,e^2\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}+\frac {3\,B\,d\,e^2\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c} \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 \sqrt {x \left (b + c x\right )} \left (A + B x\right ) \left (d + e x\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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