Optimal. Leaf size=194 \[ \frac {2 \sqrt {a^2 x^2+b^2} \left (12 a^4 c x^5+60 a^4 d x^3-3 a^2 b^2 c x^3+25 a^2 b^2 d x-4 b^4 c x\right )}{15 a^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}+\frac {2 \left (84 a^6 c x^6+420 a^6 d x^4+21 a^4 b^2 c x^4+385 a^4 b^2 d x^2-49 a^2 b^4 c x^2+35 a^2 b^4 d-8 b^6 c\right )}{105 a^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 196, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 2117, 14, 2119, 448} \begin {gather*} -\frac {b^2 d}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {d \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}-\frac {b^2 c \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{4 a^3}+\frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}{20 a^3}-\frac {b^6 c}{28 a^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}+\frac {b^4 c}{12 a^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 448
Rule 2117
Rule 2119
Rule 6742
Rubi steps
\begin {align*} \int \frac {d+c x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=c \int \frac {x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+d \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=\frac {c \operatorname {Subst}\left (\int \frac {\left (-b^2+x^2\right )^2 \left (b^2+x^2\right )}{x^{9/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a^3}+\frac {d \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}\\ &=\frac {c \operatorname {Subst}\left (\int \left (\frac {b^6}{x^{9/2}}-\frac {b^4}{x^{5/2}}-\frac {b^2}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a^3}+\frac {d \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}\\ &=-\frac {b^6 c}{28 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}+\frac {b^4 c}{12 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-\frac {b^2 d}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-\frac {b^2 c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a^3}+\frac {d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}{20 a^3}\\ \end {align*}
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Mathematica [B] time = 7.39, size = 952, normalized size = 4.91 \begin {gather*} \frac {2 \sqrt {b^2+a^2 x^2} \left (\frac {21 a^2 d \left (7 b^4+3 a x \left (7 a x+6 \sqrt {b^2+a^2 x^2}\right ) b^2+6 a^3 x^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right ) \left (b^2+2 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^5}{b^{10}+a x \left (41 a x+9 \sqrt {b^2+a^2 x^2}\right ) b^8+40 a^3 x^3 \left (7 a x+3 \sqrt {b^2+a^2 x^2}\right ) b^6+16 a^5 x^5 \left (43 a x+27 \sqrt {b^2+a^2 x^2}\right ) b^4+64 a^7 x^7 \left (11 a x+9 \sqrt {b^2+a^2 x^2}\right ) b^2+256 a^9 x^9 \left (a x+\sqrt {b^2+a^2 x^2}\right )}+\frac {5 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2 \left (-8 b^8-7 a x \left (7 a x+4 \sqrt {b^2+a^2 x^2}\right ) b^6+7 a^3 x^3 \left (a x-4 \sqrt {b^2+a^2 x^2}\right ) b^4+28 a^5 x^5 \left (4 a x+3 \sqrt {b^2+a^2 x^2}\right ) b^2+56 a^7 x^7 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right ) \left (b^2+2 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^4}{b^{12}+a x \left (61 a x+11 \sqrt {b^2+a^2 x^2}\right ) b^{10}+20 a^3 x^3 \left (31 a x+11 \sqrt {b^2+a^2 x^2}\right ) b^8+112 a^5 x^5 \left (21 a x+11 \sqrt {b^2+a^2 x^2}\right ) b^6+256 a^7 x^7 \left (16 a x+11 \sqrt {b^2+a^2 x^2}\right ) b^4+256 a^9 x^9 \left (13 a x+11 \sqrt {b^2+a^2 x^2}\right ) b^2+1024 a^{11} x^{11} \left (a x+\sqrt {b^2+a^2 x^2}\right )}-\frac {21 a^2 d \left (a x+\sqrt {b^2+a^2 x^2}\right )^2 \left (2 b^4+3 a x \left (2 a x+\sqrt {b^2+a^2 x^2}\right ) b^2+6 a^3 x^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{b^2+a x \left (a x+\sqrt {b^2+a^2 x^2}\right )}+\frac {c \left (16 b^8+14 a x \left (7 a x+4 \sqrt {b^2+a^2 x^2}\right ) b^6+7 a^3 x^3 \left (4 a x+11 \sqrt {b^2+a^2 x^2}\right ) b^4-28 a^5 x^5 \left (11 a x+6 \sqrt {b^2+a^2 x^2}\right ) b^2-280 a^7 x^7 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{b^2+a x \left (a x+\sqrt {b^2+a^2 x^2}\right )}\right )}{315 a^3 b^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 194, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b^2+a^2 x^2} \left (-4 b^4 c x+25 a^2 b^2 d x-3 a^2 b^2 c x^3+60 a^4 d x^3+12 a^4 c x^5\right )}{15 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}+\frac {2 \left (-8 b^6 c+35 a^2 b^4 d-49 a^2 b^4 c x^2+385 a^4 b^2 d x^2+21 a^4 b^2 c x^4+420 a^6 d x^4+84 a^6 c x^6\right )}{105 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 112, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (15 \, a^{4} c x^{4} + 8 \, b^{4} c - 35 \, a^{2} b^{2} d + {\left (a^{2} b^{2} c + 35 \, a^{4} d\right )} x^{2} - {\left (15 \, a^{3} c x^{3} + {\left (4 \, a b^{2} c + 35 \, a^{3} d\right )} x\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{105 \, a^{3} b^{2}} \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}{\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 \[\int \frac {c \,x^{2}+d}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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}{\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}}} \,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}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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