Optimal. Leaf size=108 \[ \frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{8 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b} \]
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Rubi [A] time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {416, 388, 217, 206} \[ \frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{8 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 416
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2}} \, dx &=\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}+\frac {\int \frac {c (4 b c-a d)+3 d (2 b c-a d) x^2}{\sqrt {a+b x^2}} \, dx}{4 b}\\ &=\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}-\frac {(3 a d (2 b c-a d)-2 b c (4 b c-a d)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^2}\\ &=\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}-\frac {(3 a d (2 b c-a d)-2 b c (4 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^2}\\ &=\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 2.49, size = 160, normalized size = 1.48 \[ \frac {x \sqrt {\frac {b x^2}{a}+1} \left (-2 b x^2 \left (c+d x^2\right )^2 \, _3F_2\left (\frac {3}{2},\frac {3}{2},2;1,\frac {9}{2};-\frac {b x^2}{a}\right )-4 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (\frac {3}{2},\frac {3}{2};\frac {9}{2};-\frac {b x^2}{a}\right )+7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \, _2F_1\left (\frac {1}{2},\frac {1}{2};\frac {7}{2};-\frac {b x^2}{a}\right )\right )}{105 a \sqrt {a+b x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 192, normalized size = 1.78 \[ \left [\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (8 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, b^{3}}, -\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (8 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 90, normalized size = 0.83 \[ \frac {1}{8} \, \sqrt {b x^{2} + a} {\left (\frac {2 \, d^{2} x^{2}}{b} + \frac {8 \, b^{2} c d - 3 \, a b d^{2}}{b^{3}}\right )} x - \frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 131, normalized size = 1.21 \[ \frac {\sqrt {b \,x^{2}+a}\, d^{2} x^{3}}{4 b}+\frac {3 a^{2} d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}-\frac {a c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}+\frac {c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {3 \sqrt {b \,x^{2}+a}\, a \,d^{2} x}{8 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, c d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 109, normalized size = 1.01 \[ \frac {\sqrt {b x^{2} + a} d^{2} x^{3}}{4 \, b} + \frac {\sqrt {b x^{2} + a} c d x}{b} - \frac {3 \, \sqrt {b x^{2} + a} a d^{2} x}{8 \, b^{2}} + \frac {c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {a c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {3 \, a^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (d\,x^2+c\right )}^2}{\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.94, size = 238, normalized size = 2.20 \[ - \frac {3 a^{\frac {3}{2}} d^{2} x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {\sqrt {a} c d x \sqrt {1 + \frac {b x^{2}}{a}}}{b} - \frac {\sqrt {a} d^{2} x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{2} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {a c d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} + c^{2} \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) + \frac {d^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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