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 = 0.183417, antiderivative size = 215, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 a c d-a e^2+4 b c e\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{8 c^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{a+b x} \]
Antiderivative was successfully verified.
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Rule 1000
Rule 810
Rule 843
Rule 621
Rule 206
Rule 724
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*}
Mathematica [A] time = 0.231853, size = 161, normalized size = 0.75 \[ -\frac{\sqrt{(a+b x)^2} \left (x^2 \left (4 a c d-a e^2+4 b c e\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right )+2 \sqrt{c} \sqrt{c+x (d x+e)} (2 a c+a e x+4 b c x)-8 b c^{3/2} \sqrt{d} x^2 \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right )\right )}{8 c^{3/2} x^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.206, size = 358, normalized size = 1.7 \begin{align*}{\frac{{\it csgn} \left ( bx+a \right ) }{8\,{c}^{2}{x}^{2}} \left ( -4\,{d}^{5/2}{c}^{3/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{2}a-2\,{d}^{5/2}\sqrt{d{x}^{2}+ex+c}{x}^{3}ae+8\,{d}^{5/2}\sqrt{d{x}^{2}+ex+c}{x}^{3}bc-4\,{d}^{3/2}{c}^{3/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{2}be+4\,{d}^{5/2}\sqrt{d{x}^{2}+ex+c}{x}^{2}ac+{d}^{{\frac{3}{2}}}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c} \right ) } \right ){x}^{2}a{e}^{2}+2\,{d}^{3/2} \left ( d{x}^{2}+ex+c \right ) ^{3/2}xae-8\,{d}^{3/2} \left ( d{x}^{2}+ex+c \right ) ^{3/2}xbc-2\,{d}^{3/2}\sqrt{d{x}^{2}+ex+c}{x}^{2}a{e}^{2}+8\,{d}^{3/2}\sqrt{d{x}^{2}+ex+c}{x}^{2}bce+8\,\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){x}^{2}b{c}^{2}{d}^{2}-4\,{d}^{3/2} \left ( d{x}^{2}+ex+c \right ) ^{3/2}ac \right ){d}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + e x + c} \sqrt{{\left (b x + a\right )}^{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.57283, size = 1670, normalized size = 7.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{2} + e x} \sqrt{\left (a + b x\right )^{2}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27092, size = 608, normalized size = 2.83 \begin{align*} -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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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