Optimal. Leaf size=267 \[ \frac {\left (b x+c x^2\right )^{3/2} \left (6 c e x (10 A c e-7 b B e+4 B c d)+10 A c e (16 c d-5 b e)+B \left (35 b^2 e^2-100 b c d e+32 c^2 d^2\right )\right )}{240 c^3}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (10 b^2 c e (A e+2 B d)-16 b c^2 d (2 A e+B d)+32 A c^3 d^2-7 b^3 B e^2\right )}{128 c^{9/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} \left (10 b^2 c e (A e+2 B d)-16 b c^2 d (2 A e+B d)+32 A c^3 d^2-7 b^3 B e^2\right )}{128 c^4}+\frac {B \left (b x+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]
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Rubi [A] time = 0.27, 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 = {832, 779, 612, 620, 206} \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 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^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 \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 206
Rule 612
Rule 620
Rule 779
Rule 832
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 ) \operatorname {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 = 0.58, size = 293, normalized size = 1.10 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {15 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\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 )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (10 b^3 c e (15 A e+30 B d+7 B e x)-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 )+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 )-105 b^4 B e^2\right )\right )}{1920 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.42, size = 361, normalized size = 1.35 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (150 A b^3 c e^2-480 A b^2 c^2 d e-100 A b^2 c^2 e^2 x+480 A b c^3 d^2+320 A b c^3 d e x+80 A b c^3 e^2 x^2+960 A c^4 d^2 x+1280 A c^4 d e x^2+480 A c^4 e^2 x^3-105 b^4 B e^2+300 b^3 B c d e+70 b^3 B c e^2 x-240 b^2 B c^2 d^2-200 b^2 B c^2 d e x-56 b^2 B c^2 e^2 x^2+160 b B c^3 d^2 x+160 b B c^3 d e x^2+48 b B c^3 e^2 x^3+640 B c^4 d^2 x^2+960 B c^4 d e x^3+384 B c^4 e^2 x^4\right )}{1920 c^4}+\frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (10 A b^4 c e^2-32 A b^3 c^2 d e+32 A b^2 c^3 d^2-7 b^5 B e^2+20 b^4 B c d e-16 b^3 B c^2 d^2\right )}{256 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 684, normalized size = 2.56 \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 (384 \, B c^{5} e^{2} x^{4} + 48 \, {\left (20 \, B c^{5} d e + {\left (B b c^{4} + 10 \, A c^{5}\right )} e^{2}\right )} x^{3} - 240 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 60 \, {\left (5 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} d e - 15 \, {\left (7 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} e^{2} + 8 \, {\left (80 \, B c^{5} d^{2} + 20 \, {\left (B b c^{4} + 8 \, A c^{5}\right )} d e - {\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} e^{2}\right )} x^{2} + 10 \, {\left (16 \, {\left (B b c^{4} + 6 \, A c^{5}\right )} d^{2} - 4 \, {\left (5 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} d e + {\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} e^{2}\right )} x\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 (384 \, B c^{5} e^{2} x^{4} + 48 \, {\left (20 \, B c^{5} d e + {\left (B b c^{4} + 10 \, A c^{5}\right )} e^{2}\right )} x^{3} - 240 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 60 \, {\left (5 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} d e - 15 \, {\left (7 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} e^{2} + 8 \, {\left (80 \, B c^{5} d^{2} + 20 \, {\left (B b c^{4} + 8 \, A c^{5}\right )} d e - {\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} e^{2}\right )} x^{2} + 10 \, {\left (16 \, {\left (B b c^{4} + 6 \, A c^{5}\right )} d^{2} - 4 \, {\left (5 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} d e + {\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} e^{2}\right )} x\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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giac [A] time = 0.22, 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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maple [B] time = 0.06, size = 671, normalized size = 2.51 \begin {gather*} -\frac {5 A \,b^{4} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {7}{2}}}+\frac {A \,b^{3} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}-\frac {A \,b^{2} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}+\frac {7 B \,b^{5} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {9}{2}}}-\frac {5 B \,b^{4} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{64 c^{\frac {7}{2}}}+\frac {B \,b^{3} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {5}{2}}}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{2} e^{2} x}{32 c^{2}}-\frac {\sqrt {c \,x^{2}+b x}\, A b d e x}{2 c}+\frac {\sqrt {c \,x^{2}+b x}\, A \,d^{2} x}{2}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{3} e^{2} x}{64 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{2} d e x}{16 c^{2}}-\frac {\sqrt {c \,x^{2}+b x}\, B b \,d^{2} x}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,e^{2} x^{2}}{5 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{3} e^{2}}{64 c^{3}}-\frac {\sqrt {c \,x^{2}+b x}\, A \,b^{2} d e}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, A b \,d^{2}}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,e^{2} x}{4 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{4} e^{2}}{128 c^{4}}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{3} d e}{32 c^{3}}-\frac {\sqrt {c \,x^{2}+b x}\, B \,b^{2} d^{2}}{8 c^{2}}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b \,e^{2} x}{40 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B d e x}{2 c}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b \,e^{2}}{24 c^{2}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A d e}{3 c}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} e^{2}}{48 c^{3}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b d e}{12 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,d^{2}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 512, normalized size = 1.92 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B e^{2} x^{2}}{5 \, c} + \frac {1}{2} \, \sqrt {c x^{2} + b x} A d^{2} x - \frac {7 \, \sqrt {c x^{2} + b x} B b^{3} e^{2} x}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b e^{2} x}{40 \, c^{2}} - \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 {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 {\sqrt {c x^{2} + b x} A b d^{2}}{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 )} \sqrt {c x^{2} + b x} b^{2} x}{32 \, c^{2}} + \frac {{\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} x}{4 \, c} - \frac {{\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b x}{4 \, c} - \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 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{3}}{64 \, c^{3}} - \frac {5 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} 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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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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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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