Optimal. Leaf size=215 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 a c d-a e^2+4 b c e\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{8 c^{3/2} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2 (a+b x)}+\frac {b \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{a+b x} \]
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Rubi [A] time = 0.18, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1000, 810, 843, 621, 206, 724} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 a c d-a e^2+4 b c e\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{8 c^{3/2} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2 (a+b x)}+\frac {b \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 843
Rule 1000
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (2 a b+2 b^2 x\right ) \sqrt {c+e x+d x^2}}{x^3} \, dx}{2 a b+2 b^2 x}\\ &=-\frac {(2 a c+(4 b c+a e) x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 c x^2 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {-b \left (4 b c e+a \left (4 c d-e^2\right )\right )-8 b^2 c d x}{x \sqrt {c+e x+d x^2}} \, dx}{4 c \left (2 a b+2 b^2 x\right )}\\ &=-\frac {(2 a c+(4 b c+a e) x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 c x^2 (a+b x)}+\frac {\left (2 b^2 d \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{2 a b+2 b^2 x}+\frac {\left (b \left (4 a c d+4 b c e-a e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{x \sqrt {c+e x+d x^2}} \, dx}{4 c \left (2 a b+2 b^2 x\right )}\\ &=-\frac {(2 a c+(4 b c+a e) x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 c x^2 (a+b x)}+\frac {\left (4 b^2 d \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 )}{2 a b+2 b^2 x}-\frac {\left (b \left (4 a c d+4 b c e-a e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {2 c+e x}{\sqrt {c+e x+d x^2}}\right )}{2 c \left (2 a b+2 b^2 x\right )}\\ &=-\frac {(2 a c+(4 b c+a e) x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 c x^2 (a+b x)}+\frac {b \sqrt {d} \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 )}{a+b x}-\frac {\left (4 a c d+4 b c e-a e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{8 c^{3/2} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 161, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (x^2 \left (4 a c d-a e^2+4 b c e\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+x (d x+e)}}\right )+2 \sqrt {c} \sqrt {c+x (d x+e)} (2 a c+a e x+4 b c x)-8 b c^{3/2} \sqrt {d} x^2 \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )\right )}{8 c^{3/2} x^2 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.81, size = 150, normalized size = 0.70 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\frac {\left (-4 a c d+a e^2-4 b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2+e x}-\sqrt {d} x}{\sqrt {c}}\right )}{4 c^{3/2}}+\frac {\sqrt {c+d x^2+e x} (-2 a c-a e x-4 b c x)}{4 c x^2}-b \sqrt {d} \log \left (-2 \sqrt {d} \sqrt {c+d x^2+e x}+2 d x+e\right )\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 693, normalized size = 3.22 \begin {gather*} \left [\frac {8 \, b c^{2} \sqrt {d} x^{2} \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 ) - {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {c} x^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} + 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - 4 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{16 \, c^{2} x^{2}}, -\frac {16 \, b c^{2} \sqrt {-d} x^{2} \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 ) + {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {c} x^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} + 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) + 4 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{16 \, c^{2} x^{2}}, \frac {4 \, b c^{2} \sqrt {d} x^{2} \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 ) + {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - 2 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{8 \, c^{2} x^{2}}, -\frac {8 \, b c^{2} \sqrt {-d} x^{2} \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 ) - {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) + 2 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{8 \, c^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 450, normalized size = 2.09 \begin {gather*} -b \sqrt {d} \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} d - \sqrt {d} e \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (4 \, a c d \mathrm {sgn}\left (b x + a\right ) + 4 \, b c e \mathrm {sgn}\left (b x + a\right ) - a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (-\frac {\sqrt {d} x - \sqrt {d x^{2} + x e + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c} c} + \frac {4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} a c d \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} b c e \mathrm {sgn}\left (b x + a\right ) + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} b c^{2} \sqrt {d} \mathrm {sgn}\left (b x + a\right ) + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} a c \sqrt {d} e \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} a c^{2} d \mathrm {sgn}\left (b x + a\right ) + {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} a e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} b c^{2} e \mathrm {sgn}\left (b x + a\right ) - 8 \, b c^{3} \sqrt {d} \mathrm {sgn}\left (b x + a\right ) + {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} a c e^{2} \mathrm {sgn}\left (b x + a\right )}{4 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} - c\right )}^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 358, normalized size = 1.67 \begin {gather*} \frac {\left (-4 a \,c^{\frac {3}{2}} d^{\frac {5}{2}} x^{2} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )+a \sqrt {c}\, d^{\frac {3}{2}} e^{2} x^{2} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )+8 b \,c^{2} d^{2} x^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-4 b \,c^{\frac {3}{2}} d^{\frac {3}{2}} e \,x^{2} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-2 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} e \,x^{3}+8 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {5}{2}} x^{3}+4 \sqrt {d \,x^{2}+e x +c}\, a c \,d^{\frac {5}{2}} x^{2}-2 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {3}{2}} e^{2} x^{2}+8 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {3}{2}} e \,x^{2}+2 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {3}{2}} e x -8 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b c \,d^{\frac {3}{2}} x -4 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a c \,d^{\frac {3}{2}}\right ) \mathrm {csgn}\left (b x +a \right )}{8 c^{2} d^{\frac {3}{2}} x^{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 \frac {\sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}}}{x^{3}}\,{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 \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^3} \,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 \frac {\sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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