Optimal. Leaf size=74 \[ \frac {2 d \sqrt {c+d x^2}}{3 \sqrt {a+b x^2} (b c-a d)^2}-\frac {\sqrt {c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {444, 45, 37} \begin {gather*} \frac {2 d \sqrt {c+d x^2}}{3 \sqrt {a+b x^2} (b c-a d)^2}-\frac {\sqrt {c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^2}}{3 (b c-a d) \left (a+b x^2\right )^{3/2}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{3 (b c-a d)}\\ &=-\frac {\sqrt {c+d x^2}}{3 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {2 d \sqrt {c+d x^2}}{3 (b c-a d)^2 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.70 \begin {gather*} \frac {\sqrt {c+d x^2} \left (3 a d-b c+2 b d x^2\right )}{3 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.81, size = 65, normalized size = 0.88 \begin {gather*} \frac {\frac {3 d \sqrt {c+d x^2}}{\sqrt {a+b x^2}}-\frac {b \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2}}}{3 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.43, size = 126, normalized size = 1.70 \begin {gather*} \frac {{\left (2 \, b d x^{2} - b c + 3 \, a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 129, normalized size = 1.74 \begin {gather*} \frac {4 \, {\left (b^{2} c - a b d - 3 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{2} d}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d \,x^{2}+c}\, \left (2 b d \,x^{2}+3 a d -b c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 137, normalized size = 1.85 \begin {gather*} \frac {\sqrt {b\,x^2+a}\,\left (\frac {x^2\,\left (3\,a\,d^2+b\,c\,d\right )}{3\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {b\,c^2-3\,a\,c\,d}{3\,b^2\,{\left (a\,d-b\,c\right )}^2}+\frac {2\,d^2\,x^4}{3\,b\,{\left (a\,d-b\,c\right )}^2}\right )}{x^4\,\sqrt {d\,x^2+c}+\frac {a^2\,\sqrt {d\,x^2+c}}{b^2}+\frac {2\,a\,x^2\,\sqrt {d\,x^2+c}}{b}} \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 {x}{\left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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