Optimal. Leaf size=164 \[ \frac {5 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 d^3}-\frac {5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 \sqrt {b} d^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {455, 52, 65,
223, 212} \begin {gather*} -\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 \sqrt {b} d^{7/2}}+\frac {5 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2}{16 d^3}-\frac {5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rule 455
Rubi steps
\begin {align*} \int \frac {x \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}-\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{12 d}\\ &=-\frac {5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{16 d^2}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 d^3}-\frac {5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}-\frac {\left (5 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{32 d^3}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 d^3}-\frac {5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}-\frac {\left (5 (b c-a d)^3\right ) \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 )}{16 b d^3}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 d^3}-\frac {5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}-\frac {\left (5 (b c-a d)^3\right ) \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 )}{16 b d^3}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 d^3}-\frac {5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 d}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 \sqrt {b} d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.68, size = 137, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (33 a^2 d^2+2 a b d \left (-20 c+13 d x^2\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )}{48 d^3}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{16 \sqrt {b} d^{7/2}} \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. \(451\) vs.
\(2(132)=264\).
time = 0.13, size = 452, normalized size = 2.76
method | result | size |
risch | \(\frac {\left (8 b^{2} x^{4} d^{2}+26 a b \,d^{2} x^{2}-10 b^{2} c d \,x^{2}+33 a^{2} d^{2}-40 a b c d +15 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{48 d^{3}}+\frac {\left (\frac {5 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{3}}{32 \sqrt {b d}}-\frac {15 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} b c}{32 d \sqrt {b d}}+\frac {15 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a \,b^{2} c^{2}}{32 d^{2} \sqrt {b d}}-\frac {5 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) b^{3} c^{3}}{32 d^{3} \sqrt {b d}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(358\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (16 b^{2} d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+52 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x^{2} a b \,d^{2} \sqrt {b d}-20 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x^{2} c \,b^{2} d \sqrt {b d}+15 d^{3} \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3}-45 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c b \,d^{2}+45 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,c^{2} b^{2} d -15 b^{3} \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) c^{3}+66 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a^{2} d^{2} \sqrt {b d}-80 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a c b d \sqrt {b d}+30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, c^{2} b^{2} \sqrt {b d}\right )}{96 d^{3} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\) | \(452\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{2} x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{6 d}+\frac {13 b \,x^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{24 d}-\frac {5 b^{2} x^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{24 d^{2}}+\frac {11 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2}}{16 d}-\frac {5 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a c}{6 d^{2}}+\frac {5 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{2}}{16 d^{3}}+\frac {5 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{3}}{32 \sqrt {b d}}-\frac {15 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} b c}{32 d \sqrt {b d}}+\frac {15 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a \,b^{2} c^{2}}{32 d^{2} \sqrt {b d}}-\frac {5 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) b^{3} c^{3}}{32 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(478\) |
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 [A]
time = 1.33, size = 440, normalized size = 2.68 \begin {gather*} \left [-\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{192 \, b d^{4}}, \frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{96 \, b d^{4}}\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 \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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Giac [A]
time = 1.04, size = 210, normalized size = 1.28 \begin {gather*} \frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (\frac {4 \, {\left (b x^{2} + a\right )}}{b d} - \frac {5 \, {\left (b c d^{3} - a d^{4}\right )}}{b d^{5}}\right )} + \frac {15 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}}{b d^{5}}\right )} + \frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 |}\right )}{\sqrt {b d} d^{3}}\right )} b}{48 \, {\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\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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