Optimal. Leaf size=83 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{b^{3/2} \sqrt {d}}+\frac {a \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)} \]
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Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {446, 78, 63, 217, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{b^{3/2} \sqrt {d}}+\frac {a \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {a \sqrt {c+d x^2}}{b (b c-a d) \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b}\\ &=\frac {a \sqrt {c+d x^2}}{b (b c-a d) \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{b^2}\\ &=\frac {a \sqrt {c+d x^2}}{b (b c-a d) \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{b^2}\\ &=\frac {a \sqrt {c+d x^2}}{b (b c-a d) \sqrt {a+b x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{b^{3/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 118, normalized size = 1.42 \begin {gather*} \frac {\frac {a b \left (c+d x^2\right )}{\sqrt {a+b x^2} (b c-a d)}+\frac {\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {d}}}{b^2 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.75, size = 83, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{b^{3/2} \sqrt {d}}+\frac {a \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.44, size = 367, normalized size = 4.42 \begin {gather*} \left [\frac {4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a b d + {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right )}{4 \, {\left (a b^{3} c d - a^{2} b^{2} d^{2} + {\left (b^{4} c d - a b^{3} d^{2}\right )} x^{2}\right )}}, \frac {2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a b d - {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (a b^{3} c d - a^{2} b^{2} d^{2} + {\left (b^{4} c d - a b^{3} d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 135, normalized size = 1.63 \begin {gather*} \frac {\frac {4 \, \sqrt {b d} a b}{{\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 )} {\left | b \right |}} - \frac {\sqrt {b d} \log \left ({\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 )}{d {\left | b \right |}}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 320, normalized size = 3.86 \begin {gather*} \frac {\left (a b d \,x^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-b^{2} c \,x^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+a^{2} d \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-a b c \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-2 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a \right ) \sqrt {d \,x^{2}+c}}{2 \sqrt {b \,x^{2}+a}\, \sqrt {b d}\, \left (a d -b c \right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, b} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,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 {x^{3}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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