Optimal. Leaf size=317 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt {c+d x^2+e x} \left (2 a d \left (4 c d-5 e^2\right )-b \left (12 c d e-7 e^3\right )\right )}{128 d^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} \left (-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right )}{240 d^3 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{256 d^{9/2} (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{5 d (a+b x)} \]
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Rubi [A] time = 0.33, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1000, 832, 779, 612, 621, 206} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} \left (-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right )}{240 d^3 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt {c+d x^2+e x} \left (2 a d \left (4 c d-5 e^2\right )-b \left (12 c d e-7 e^3\right )\right )}{128 d^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{256 d^{9/2} (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{5 d (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rule 1000
Rubi steps
\begin {align*} \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (2 a b+2 b^2 x\right ) \sqrt {c+e x+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (-4 b^2 c+b (10 a d-7 b e) x\right ) \sqrt {c+e x+d x^2} \, dx}{5 d \left (2 a b+2 b^2 x\right )}\\ &=\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}+\frac {\left (\left (16 b^2 c d e-2 b c d (10 a d-7 b e)+\frac {5}{2} b e^2 (10 a d-7 b e)\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+e x+d x^2} \, dx}{40 d^3 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}+\frac {\left (\left (4 c d-e^2\right ) \left (16 b^2 c d e-2 b c d (10 a d-7 b e)+\frac {5}{2} b e^2 (10 a d-7 b e)\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{320 d^4 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}+\frac {\left (\left (4 c d-e^2\right ) \left (16 b^2 c d e-2 b c d (10 a d-7 b e)+\frac {5}{2} b e^2 (10 a d-7 b e)\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{160 d^4 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}-\frac {\left (4 c d-e^2\right ) \left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{256 d^{9/2} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 198, normalized size = 0.62 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-\frac {5 \left (2 a d \left (4 c d-5 e^2\right )+b \left (7 e^3-12 c d e\right )\right ) \left (\left (4 c d-e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )+2 \sqrt {d} (2 d x+e) \sqrt {c+x (d x+e)}\right )}{256 d^{7/2}}+\frac {(c+x (d x+e))^{3/2} (10 a d (6 d x-5 e)-32 b c d+7 b e (5 e-6 d x))}{48 d^2}+b x^2 (c+x (d x+e))^{3/2}\right )}{5 d (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.00, size = 263, normalized size = 0.83 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\frac {\left (32 a c^2 d^3-48 a c d^2 e^2+10 a d e^4-48 b c^2 d^2 e+40 b c d e^3-7 b e^5\right ) \log \left (-2 \sqrt {d} \sqrt {c+d x^2+e x}+2 d x+e\right )}{256 d^{9/2}}+\frac {\sqrt {c+d x^2+e x} \left (240 a c d^3 x-520 a c d^2 e+480 a d^4 x^3+80 a d^3 e x^2-100 a d^2 e^2 x+150 a d e^3-256 b c^2 d^2+128 b c d^3 x^2-232 b c d^2 e x+460 b c d e^2+384 b d^4 x^4+48 b d^3 e x^3-56 b d^2 e^2 x^2+70 b d e^3 x-105 b e^4\right )}{1920 d^4}\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 517, normalized size = 1.63 \begin {gather*} \left [-\frac {15 \, {\left (32 \, a c^{2} d^{3} - 48 \, b c^{2} d^{2} e - 48 \, a c d^{2} e^{2} + 40 \, b c d e^{3} + 10 \, a d e^{4} - 7 \, b e^{5}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) - 4 \, {\left (384 \, b d^{5} x^{4} - 256 \, b c^{2} d^{3} - 520 \, a c d^{3} e + 460 \, b c d^{2} e^{2} + 150 \, a d^{2} e^{3} - 105 \, b d e^{4} + 48 \, {\left (10 \, a d^{5} + b d^{4} e\right )} x^{3} + 8 \, {\left (16 \, b c d^{4} + 10 \, a d^{4} e - 7 \, b d^{3} e^{2}\right )} x^{2} + 2 \, {\left (120 \, a c d^{4} - 116 \, b c d^{3} e - 50 \, a d^{3} e^{2} + 35 \, b d^{2} e^{3}\right )} x\right )} \sqrt {d x^{2} + e x + c}}{7680 \, d^{5}}, \frac {15 \, {\left (32 \, a c^{2} d^{3} - 48 \, b c^{2} d^{2} e - 48 \, a c d^{2} e^{2} + 40 \, b c d e^{3} + 10 \, a d e^{4} - 7 \, b e^{5}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (384 \, b d^{5} x^{4} - 256 \, b c^{2} d^{3} - 520 \, a c d^{3} e + 460 \, b c d^{2} e^{2} + 150 \, a d^{2} e^{3} - 105 \, b d e^{4} + 48 \, {\left (10 \, a d^{5} + b d^{4} e\right )} x^{3} + 8 \, {\left (16 \, b c d^{4} + 10 \, a d^{4} e - 7 \, b d^{3} e^{2}\right )} x^{2} + 2 \, {\left (120 \, a c d^{4} - 116 \, b c d^{3} e - 50 \, a d^{3} e^{2} + 35 \, b d^{2} e^{3}\right )} x\right )} \sqrt {d x^{2} + e x + c}}{3840 \, d^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 368, normalized size = 1.16 \begin {gather*} \frac {1}{1920} \, \sqrt {d x^{2} + x e + c} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, b x \mathrm {sgn}\left (b x + a\right ) + \frac {10 \, a d^{4} \mathrm {sgn}\left (b x + a\right ) + b d^{3} e \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} x + \frac {16 \, b c d^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a d^{3} e \mathrm {sgn}\left (b x + a\right ) - 7 \, b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} x + \frac {120 \, a c d^{3} \mathrm {sgn}\left (b x + a\right ) - 116 \, b c d^{2} e \mathrm {sgn}\left (b x + a\right ) - 50 \, a d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, b d e^{3} \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} x - \frac {256 \, b c^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 520 \, a c d^{2} e \mathrm {sgn}\left (b x + a\right ) - 460 \, b c d e^{2} \mathrm {sgn}\left (b x + a\right ) - 150 \, a d e^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, b e^{4} \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} + \frac {{\left (32 \, a c^{2} d^{3} \mathrm {sgn}\left (b x + a\right ) - 48 \, b c^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 48 \, a c d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 40 \, b c d e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a d e^{4} \mathrm {sgn}\left (b x + a\right ) - 7 \, b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} \sqrt {d} - e \right |}\right )}{256 \, d^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 530, normalized size = 1.67 \begin {gather*} \frac {\left (-480 a \,c^{2} d^{4} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+720 a c \,d^{3} e^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-150 a \,d^{2} e^{4} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+720 b \,c^{2} d^{3} e \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-600 b c \,d^{2} e^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+105 b d \,e^{5} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-480 \sqrt {d \,x^{2}+e x +c}\, a c \,d^{\frac {9}{2}} x +600 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {7}{2}} e^{2} x +720 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {7}{2}} e x +768 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {9}{2}} x^{2}-420 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {5}{2}} e^{3} x -240 \sqrt {d \,x^{2}+e x +c}\, a c \,d^{\frac {7}{2}} e +960 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {9}{2}} x +300 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} e^{3}+360 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {5}{2}} e^{2}-672 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {7}{2}} e x -210 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {3}{2}} e^{4}-800 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {7}{2}} e -512 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b c \,d^{\frac {7}{2}}+560 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {5}{2}} e^{2}\right ) \mathrm {csgn}\left (b x +a \right )}{3840 d^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}} x^{2}\,{d x} \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 x^2\,\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c} \,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 x^{2} \sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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