Optimal. Leaf size=199 \[ a d \text {RootSum}\left [\text {$\#$1}^8 c-4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^7 (-c)+2 \text {$\#$1}^3 a^2 d+\text {$\#$1}^3 b^2 c}\& \right ]-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a} \]
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Rubi [B] time = 1.29, antiderivative size = 453, normalized size of antiderivative = 2.28, number of steps used = 23, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6725, 2117, 14, 2119, 1628, 828, 826, 1166, 208, 205} \begin {gather*} -\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}-\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 205
Rule 208
Rule 826
Rule 828
Rule 1166
Rule 1628
Rule 2117
Rule 2119
Rule 6725
Rubi steps
\begin {align*} \int \frac {d+c x^2}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 d}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=(2 d) \int \frac {1}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+\int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}+(2 d) \int \left (-\frac {1}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {1}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-\sqrt {d} \int \frac {1}{\left (\sqrt {d}-\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx-\sqrt {d} \int \frac {1}{\left (\sqrt {d}+\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {d} \operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {c} x^{3/2}}+\frac {2 \left (b^2 \sqrt {c}+a \sqrt {d} x\right )}{\sqrt {c} x^{3/2} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {d} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {c} x^{3/2}}+\frac {2 \left (b^2 \sqrt {c}-a \sqrt {d} x\right )}{\sqrt {c} x^{3/2} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 \sqrt {c}+a \sqrt {d} x}{x^{3/2} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{\sqrt {c}}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 \sqrt {c}-a \sqrt {d} x}{x^{3/2} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{\sqrt {c}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}+b^2 c x}{\sqrt {x} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 c}+\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}-b^2 c x}{\sqrt {x} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (4 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}+b^2 c x^2}{b^2 \sqrt {c}+2 a \sqrt {d} x^2-\sqrt {c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}+\frac {\left (4 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}-b^2 c x^2}{-b^2 \sqrt {c}+2 a \sqrt {d} x^2+\sqrt {c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {d}-\sqrt {b^2 c+a^2 d}-\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )-\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {d}+\sqrt {b^2 c+a^2 d}-\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )-\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {d}-\sqrt {b^2 c+a^2 d}+\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )-\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {d}+\sqrt {b^2 c+a^2 d}+\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}\right )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}-\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\right )}{\sqrt [4]{c} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}\right )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\right )}{\sqrt [4]{c} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 194, normalized size = 0.97 \begin {gather*} \frac {a^2 (-d) \text {RootSum}\left [\text {$\#$1}^8 b^4 c-4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+c\&,\frac {\text {$\#$1}^4 b^2 \log \left (\frac {1}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\text {$\#$1}\right )+\log \left (\frac {1}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\text {$\#$1}\right )}{\text {$\#$1}^5 b^4 c-2 \text {$\#$1} a^2 d-\text {$\#$1} b^2 c}\&\right ]+\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\frac {b^2}{3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}}{a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 199, normalized size = 1.00 \begin {gather*} -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3+2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{2} + d}{{\left (c x^{2} - d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{2}+d}{\left (c \,x^{2}-d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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 {c x^{2} + d}{{\left (c x^{2} - d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{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 -\frac {c\,x^2+d}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (d-c\,x^2\right )} \,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 {c x^{2} + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} - d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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