Optimal. Leaf size=198 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) (2 a d-b e) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{16 d^{5/2} (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) (2 a d-b e) \sqrt {c+d x^2+e x}}{8 d^2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{3 d (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {969, 640, 612, 621, 206} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) (2 a d-b e) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{16 d^{5/2} (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) (2 a d-b e) \sqrt {c+d x^2+e x}}{8 d^2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{3 d (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 969
Rubi steps
\begin {align*} \int \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 \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 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {\left (b (2 a d-b e) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+e x+d x^2} \, dx}{d \left (2 a b+2 b^2 x\right )}\\ &=\frac {(2 a d-b e) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{8 d^2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {\left (b (2 a d-b e) \left (4 c d-e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{8 d^2 \left (2 a b+2 b^2 x\right )}\\ &=\frac {(2 a d-b e) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{8 d^2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {\left (b (2 a d-b e) \left (4 c d-e^2\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 )}{4 d^2 \left (2 a b+2 b^2 x\right )}\\ &=\frac {(2 a d-b e) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{8 d^2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {(2 a d-b e) \left (4 c d-e^2\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 )}{16 d^{5/2} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 134, normalized size = 0.68 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (2 \sqrt {d} \sqrt {c+x (d x+e)} \left (6 a d (2 d x+e)+b \left (8 c d+8 d^2 x^2+2 d e x-3 e^2\right )\right )+3 \left (4 c d-e^2\right ) (2 a d-b e) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )\right )}{48 d^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 150, normalized size = 0.76 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\frac {\sqrt {c+d x^2+e x} \left (12 a d^2 x+6 a d e+8 b c d+8 b d^2 x^2+2 b d e x-3 b e^2\right )}{24 d^2}+\frac {\left (-8 a c d^2+2 a d e^2+4 b c d e-b e^3\right ) \log \left (-2 d^{5/2} \sqrt {c+d x^2+e x}+2 d^3 x+d^2 e\right )}{16 d^{5/2}}\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 287, normalized size = 1.45 \begin {gather*} \left [\frac {3 \, {\left (8 \, a c d^{2} - 4 \, b c d e - 2 \, a d e^{2} + b e^{3}\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 (8 \, b d^{3} x^{2} + 8 \, b c d^{2} + 6 \, a d^{2} e - 3 \, b d e^{2} + 2 \, {\left (6 \, a d^{3} + b d^{2} e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{96 \, d^{3}}, -\frac {3 \, {\left (8 \, a c d^{2} - 4 \, b c d e - 2 \, a d e^{2} + b e^{3}\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 (8 \, b d^{3} x^{2} + 8 \, b c d^{2} + 6 \, a d^{2} e - 3 \, b d e^{2} + 2 \, {\left (6 \, a d^{3} + b d^{2} e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{48 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 185, normalized size = 0.93 \begin {gather*} \frac {1}{24} \, \sqrt {d x^{2} + x e + c} {\left (2 \, {\left (4 \, b x \mathrm {sgn}\left (b x + a\right ) + \frac {6 \, a d^{2} \mathrm {sgn}\left (b x + a\right ) + b d e \mathrm {sgn}\left (b x + a\right )}{d^{2}}\right )} x + \frac {8 \, b c d \mathrm {sgn}\left (b x + a\right ) + 6 \, a d e \mathrm {sgn}\left (b x + a\right ) - 3 \, b e^{2} \mathrm {sgn}\left (b x + a\right )}{d^{2}}\right )} - \frac {{\left (8 \, a c d^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, b c d e \mathrm {sgn}\left (b x + a\right ) - 2 \, a d e^{2} \mathrm {sgn}\left (b x + a\right ) + b e^{3} \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 )}{16 \, d^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 257, normalized size = 1.30 \begin {gather*} \frac {\left (24 a c \,d^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-6 a \,d^{2} e^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-12 b c \,d^{2} e \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+3 b d \,e^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+24 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {7}{2}} x -12 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {5}{2}} e x +12 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} e -6 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {3}{2}} e^{2}+16 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {5}{2}}\right ) \mathrm {csgn}\left (b x +a \right )}{48 d^{\frac {7}{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}}\,{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 = -0.01 \begin {gather*} \int \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 \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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