Optimal. Leaf size=129 \[ -\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d}-\frac {(b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 b^{5/2} d^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 91, 81, 65,
223, 212} \begin {gather*} -\frac {a^2 \sqrt {c+d x^2}}{b^2 \sqrt {a+b x^2} (b c-a d)}-\frac {(3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 91
Rule 212
Rule 223
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} a (b c-a d)+\frac {1}{2} b (b c-a d) x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{b^2 (b c-a d)}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d}-\frac {(b c+3 a d) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 b^2 d}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d}-\frac {(b c+3 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{2 b^3 d}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d}-\frac {(b c+3 a d) \text {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 )}{2 b^3 d}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{b^2 (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b^2 d}-\frac {(b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.89, size = 153, normalized size = 1.19 \begin {gather*} \frac {\sqrt {b} \sqrt {d} \sqrt {c+d x^2} \left (-3 a^2 d+b^2 c x^2+a b \left (c-d x^2\right )\right )-\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 b^{5/2} d^{3/2} (b c-a d) \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(510\) vs.
\(2(105)=210\).
time = 0.16, size = 511, normalized size = 3.96
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 b^{2} d}-\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a}{4 b^{2} \sqrt {b d}}-\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c}{4 b d \sqrt {b d}}-\frac {a^{2} \sqrt {b d \left (x^{2}+\frac {a}{b}\right )^{2}+\left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}}{b^{3} \left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(250\) |
risch | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{2 b^{2} d}+\frac {\left (-\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a}{4 b^{2} \sqrt {b d}}-\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c}{4 b d \sqrt {b d}}+\frac {d \,a^{2} x^{2}}{b^{2} \left (a d -b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}+\frac {a^{2} c}{b^{2} \left (a d -b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(269\) |
default | \(-\frac {\left (3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,d^{2} x^{2}-2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c d \,x^{2}-\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{2}-2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c \,x^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{2}-2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c d -\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2}-6 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a^{2} d +2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b c \right ) \sqrt {d \,x^{2}+c}}{4 b^{2} \sqrt {b \,x^{2}+a}\, d \sqrt {b d}\, \left (a d -b c \right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\) | \(511\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs.
\(2 (105) = 210\).
time = 1.00, size = 498, normalized size = 3.86 \begin {gather*} \left [\frac {{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\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^{2} c d - 3 \, a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2}\right )}}, \frac {{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\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^{2} c d - 3 \, a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2}\right )}}\right ] \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^{5}}{\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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Giac [A]
time = 0.90, size = 192, normalized size = 1.49 \begin {gather*} -\frac {2 \, \sqrt {b d} a^{2}}{{\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 )} b {\left | b \right |}} + \frac {\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left | b \right |}}{2 \, b^{4} d} + \frac {{\left (\sqrt {b d} b c + 3 \, \sqrt {b d} a d\right )} \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 )}{4 \, b^{2} d^{2} {\left | b \right |}} \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^5}{{\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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