Optimal. Leaf size=196 \[ a d \text {RootSum}\left [\text {$\#$1}^8 c-4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 (-c)+2 \text {$\#$1} a^2 d+\text {$\#$1} b^2 c}\& \right ]-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \]
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Rubi [B] time = 1.22, antiderivative size = 454, normalized size of antiderivative = 2.32, number of steps used = 21, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6725, 2117, 14, 2119, 1628, 826, 1166, 208, 205} \begin {gather*} \frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \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 )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \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 )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \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 )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \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 )}{c^{3/4}}-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 205
Rule 208
Rule 826
Rule 1166
Rule 1628
Rule 2117
Rule 2119
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx &=\int \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}+\frac {2 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2}\right ) \, dx\\ &=(2 d) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx+\int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}+(2 d) \int \left (-\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {c} x\right )}-\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {c} x\right )}\right ) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-\sqrt {d} \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {d}-\sqrt {c} x} \, dx-\sqrt {d} \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {d}+\sqrt {c} x} \, dx\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\sqrt {d} \operatorname {Subst}\left (\int \frac {b^2+x^2}{\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 )-\sqrt {d} \operatorname {Subst}\left (\int \frac {b^2+x^2}{\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 )\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\sqrt {d} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {c} \sqrt {x}}+\frac {2 \left (b^2 \sqrt {c}+a \sqrt {d} x\right )}{\sqrt {c} \sqrt {x} \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} \sqrt {x}}+\frac {2 \left (b^2 \sqrt {c}-a \sqrt {d} x\right )}{\sqrt {c} \sqrt {x} \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}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 \sqrt {c}+a \sqrt {d} 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 )}{\sqrt {c}}-\frac {\left (2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 \sqrt {c}-a \sqrt {d} 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 )}{\sqrt {c}}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (4 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 \sqrt {c}+a \sqrt {d} 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 )}{\sqrt {c}}-\frac {\left (4 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {b^2 \sqrt {c}-a \sqrt {d} 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 )}{\sqrt {c}}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (2 \sqrt {d} \left (a \sqrt {d}-\sqrt {b^2 c+a^2 d}\right )\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 )}{\sqrt {c}}+\frac {\left (2 \sqrt {d} \left (a \sqrt {d}-\sqrt {b^2 c+a^2 d}\right )\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 )}{\sqrt {c}}-\frac {\left (2 \sqrt {d} \left (a \sqrt {d}+\sqrt {b^2 c+a^2 d}\right )\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 )}{\sqrt {c}}+\frac {\left (2 \sqrt {d} \left (a \sqrt {d}+\sqrt {b^2 c+a^2 d}\right )\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 )}{\sqrt {c}}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 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 )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 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 )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 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 )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 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 )}{c^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 190, normalized size = 0.97 \begin {gather*} a d \text {RootSum}\left [\text {$\#$1}^8 c-4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\&,\frac {\text {$\#$1}^4 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 c-2 \text {$\#$1} a^2 d-\text {$\#$1} b^2 c}\&\right ]-\frac {2 \left (b^2-a x \left (\sqrt {a^2 x^2+b^2}+a x\right )\right )}{3 a \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 196, normalized size = 1.00 \begin {gather*} -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}+2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\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 {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{c x^{2} - d}\,{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 \[\int \frac {\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{c \,x^{2}-d}\, dx\]
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 {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{c x^{2} - d}\,{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 {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )}{d-c\,x^2} \,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 {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}{c x^{2} - d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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