Optimal. Leaf size=267 \[ \frac {\left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (10 A c e (16 c d-5 b e)+B \left (32 c^2 d^2-100 b c d e+35 b^2 e^2\right )+6 c e (4 B c d-7 b B e+10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {b^2 \left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {846, 793, 626,
634, 212} \begin {gather*} \frac {\left (b x+c x^2\right )^{3/2} \left (6 c e x (10 A c e-7 b B e+4 B c d)+10 A c e (16 c d-5 b e)+B \left (35 b^2 e^2-100 b c d e+32 c^2 d^2\right )\right )}{240 c^3}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (10 b^2 c e (A e+2 B d)-16 b c^2 d (2 A e+B d)+32 A c^3 d^2-7 b^3 B e^2\right )}{128 c^{9/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} \left (10 b^2 c e (A e+2 B d)-16 b c^2 d (2 A e+B d)+32 A c^3 d^2-7 b^3 B e^2\right )}{128 c^4}+\frac {B \left (b x+c x^2\right )^{3/2} (d+e x)^2}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 634
Rule 793
Rule 846
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \sqrt {b x+c x^2} \, dx &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\int (d+e x) \left (-\frac {1}{2} (3 b B-10 A c) d+\frac {1}{2} (4 B c d-7 b B e+10 A c e) x\right ) \sqrt {b x+c x^2} \, dx}{5 c}\\ &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (10 A c e (16 c d-5 b e)+B \left (32 c^2 d^2-100 b c d e+35 b^2 e^2\right )+6 c e (4 B c d-7 b B e+10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}+\frac {\left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) \int \sqrt {b x+c x^2} \, dx}{32 c^3}\\ &=\frac {\left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (10 A c e (16 c d-5 b e)+B \left (32 c^2 d^2-100 b c d e+35 b^2 e^2\right )+6 c e (4 B c d-7 b B e+10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2 \left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^4}\\ &=\frac {\left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (10 A c e (16 c d-5 b e)+B \left (32 c^2 d^2-100 b c d e+35 b^2 e^2\right )+6 c e (4 B c d-7 b B e+10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2 \left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^4}\\ &=\frac {\left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (10 A c e (16 c d-5 b e)+B \left (32 c^2 d^2-100 b c d e+35 b^2 e^2\right )+6 c e (4 B c d-7 b B e+10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{240 c^3}-\frac {b^2 \left (32 A c^3 d^2-7 b^3 B e^2+10 b^2 c e (2 B d+A e)-16 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 294, normalized size = 1.10 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-105 b^4 B e^2+10 b^3 c e (30 B d+15 A e+7 B e x)+16 b c^3 \left (5 A \left (6 d^2+4 d e x+e^2 x^2\right )+B x \left (10 d^2+10 d e x+3 e^2 x^2\right )\right )+32 c^4 x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )-4 b^2 c^2 \left (5 A e (24 d+5 e x)+2 B \left (30 d^2+25 d e x+7 e^2 x^2\right )\right )\right )-\frac {15 b^2 \left (-32 A c^3 d^2+7 b^3 B e^2-10 b^2 c e (2 B d+A e)+16 b c^2 d (B d+2 A e)\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{1920 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 399, normalized size = 1.49
method | result | size |
default | \(B \,e^{2} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )}{10 c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )+\left (2 A d e +B \,d^{2}\right ) \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )+A \,d^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )\) | \(399\) |
risch | \(\frac {\left (384 B \,c^{4} e^{2} x^{4}+480 A \,c^{4} e^{2} x^{3}+48 B b \,c^{3} e^{2} x^{3}+960 B \,c^{4} d e \,x^{3}+80 A b \,c^{3} e^{2} x^{2}+1280 A \,c^{4} d e \,x^{2}-56 B \,b^{2} c^{2} e^{2} x^{2}+160 B b \,c^{3} d e \,x^{2}+640 B \,c^{4} d^{2} x^{2}-100 A \,b^{2} c^{2} e^{2} x +320 A b \,c^{3} d e x +960 A \,c^{4} d^{2} x +70 B \,b^{3} c \,e^{2} x -200 B \,b^{2} c^{2} d e x +160 B b \,c^{3} d^{2} x +150 A \,b^{3} c \,e^{2}-480 A \,b^{2} c^{2} d e +480 A b \,c^{3} d^{2}-105 B \,b^{4} e^{2}+300 B \,b^{3} c d e -240 B \,b^{2} c^{2} d^{2}\right ) x \left (c x +b \right )}{1920 c^{4} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A \,e^{2}}{128 c^{\frac {7}{2}}}+\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A d e}{8 c^{\frac {5}{2}}}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A \,d^{2}}{8 c^{\frac {3}{2}}}+\frac {7 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B \,e^{2}}{256 c^{\frac {9}{2}}}-\frac {5 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B d e}{64 c^{\frac {7}{2}}}+\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B \,d^{2}}{16 c^{\frac {5}{2}}}\) | \(476\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 510, normalized size = 1.91 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + b x} A d^{2} x + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B x^{2} e^{2}}{5 \, c} - \frac {A b^{2} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x} A b d^{2}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{3} x e^{2}}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b x e^{2}}{40 \, c^{2}} + \frac {7 \, B b^{5} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {9}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} {\left (2 \, B d e + A e^{2}\right )} b^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (2 \, B d e + A e^{2}\right )} x}{4 \, c} - \frac {{\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b x}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{4} e^{2}}{128 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} e^{2}}{48 \, c^{3}} - \frac {5 \, {\left (2 \, B d e + A 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^{2} + 2 \, A d 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 {5 \, \sqrt {c x^{2} + b x} {\left (2 \, B d e + A e^{2}\right )} b^{3}}{64 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (2 \, B d e + A e^{2}\right )} b}{24 \, c^{2}} - \frac {{\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{2}} + \frac {{\left (B d^{2} + 2 \, A d 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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Fricas [A]
time = 3.70, size = 666, normalized size = 2.49 \begin {gather*} \left [-\frac {15 \, {\left (16 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} - 4 \, {\left (5 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} d e + {\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (640 \, B c^{5} d^{2} x^{2} + 160 \, {\left (B b c^{4} + 6 \, A c^{5}\right )} d^{2} x - 240 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + {\left (384 \, B c^{5} x^{4} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} + 48 \, {\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 8 \, {\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} e^{2} + 20 \, {\left (48 \, B c^{5} d x^{3} + 8 \, {\left (B b c^{4} + 8 \, A c^{5}\right )} d x^{2} - 2 \, {\left (5 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} d x + 3 \, {\left (5 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x}}{3840 \, c^{5}}, -\frac {15 \, {\left (16 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} - 4 \, {\left (5 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} d e + {\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (640 \, B c^{5} d^{2} x^{2} + 160 \, {\left (B b c^{4} + 6 \, A c^{5}\right )} d^{2} x - 240 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + {\left (384 \, B c^{5} x^{4} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} + 48 \, {\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 8 \, {\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} e^{2} + 20 \, {\left (48 \, B c^{5} d x^{3} + 8 \, {\left (B b c^{4} + 8 \, A c^{5}\right )} d x^{2} - 2 \, {\left (5 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} d x + 3 \, {\left (5 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x}}{1920 \, c^{5}}\right ] \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 )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 349, normalized size = 1.31 \begin {gather*} \frac {1}{1920} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, B x e^{2} + \frac {20 \, B c^{4} d e + B b c^{3} e^{2} + 10 \, A c^{4} e^{2}}{c^{4}}\right )} x + \frac {80 \, B c^{4} d^{2} + 20 \, B b c^{3} d e + 160 \, A c^{4} d e - 7 \, B b^{2} c^{2} e^{2} + 10 \, A b c^{3} e^{2}}{c^{4}}\right )} x + \frac {5 \, {\left (16 \, B b c^{3} d^{2} + 96 \, A c^{4} d^{2} - 20 \, B b^{2} c^{2} d e + 32 \, A b c^{3} d e + 7 \, B b^{3} c e^{2} - 10 \, A b^{2} c^{2} e^{2}\right )}}{c^{4}}\right )} x - \frac {15 \, {\left (16 \, B b^{2} c^{2} d^{2} - 32 \, A b c^{3} d^{2} - 20 \, B b^{3} c d e + 32 \, A b^{2} c^{2} d e + 7 \, B b^{4} e^{2} - 10 \, A b^{3} c e^{2}\right )}}{c^{4}}\right )} - \frac {{\left (16 \, B b^{3} c^{2} d^{2} - 32 \, A b^{2} c^{3} d^{2} - 20 \, B b^{4} c d e + 32 \, A b^{3} c^{2} d e + 7 \, B b^{5} e^{2} - 10 \, A b^{4} c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.03, size = 537, normalized size = 2.01 \begin {gather*} A\,d^2\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )+\frac {A\,e^2\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,A\,b\,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 {7\,B\,b\,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 {B\,e^2\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}-\frac {A\,b^2\,d^2\,\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^2\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {B\,d^2\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}+\frac {B\,d\,e\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{2\,c}-\frac {5\,B\,b\,d\,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 )}{4\,c}+\frac {A\,b^3\,d\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{8\,c^{5/2}}+\frac {A\,d\,e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{12\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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