Optimal. Leaf size=362 \[ -\frac {(2 b+a c) d^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^2 (b+a c)}-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c x^3}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^2 (b+a c) x}+\frac {(2 b+a c) d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.29, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986,
486, 597, 545, 429, 506, 422} \begin {gather*} \frac {d^{3/2} (a c+2 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {a d^{3/2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {d^2 x (a c+2 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^2 (a c+b)}+\frac {d (a c+2 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^2 x (a c+b)}-\frac {\left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 486
Rule 506
Rule 545
Rule 597
Rule 1985
Rule 1986
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a \left (c+d x^2\right )}}{x^4 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{x^4 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {-(2 b+a c) d-a d^2 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {a c (b+a c) d^2+a (2 b+a c) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (a d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (a (2 b+a c) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(2 b+a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {a d^{3/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left ((2 b+a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(2 b+a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d^{3/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {a d^{3/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.66, size = 314, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {\frac {a d}{b+a c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} \left (c+d x^2\right ) \left (b^2 \left (c-2 d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+2 a b \left (c^2-c d x^2-d^2 x^4\right )\right )-i a c (2 b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )+i a b c d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{3 a c^2 d x^3 \left (b+a \left (c+d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 571, normalized size = 1.58
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (-a c d \,x^{2}-2 b d \,x^{2}+c^{2} a +b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 c^{2} x^{3} \left (a c +b \right )}-\frac {a \,d^{2} \left (-\frac {2 \left (a c d +2 b d \right ) \left (c^{2} a +b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \left (2 a c d +2 b d \right )}+\frac {c^{2} a \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{3 c^{2} \left (a c +b \right ) \left (a d \,x^{2}+a c +b \right )}\) | \(569\) |
default | \(-\frac {\left (-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{3} x^{6}-2 \sqrt {-\frac {a d}{a c +b}}\, a b \,d^{3} x^{6}+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{2} d^{2} x^{3}-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d^{2} x^{4}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}+2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}-4 \sqrt {-\frac {a d}{a c +b}}\, a b c \,d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} d \,x^{2}-2 \sqrt {-\frac {a d}{a c +b}}\, b^{2} d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{4}-\sqrt {-\frac {a d}{a c +b}}\, b^{2} c d \,x^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{3}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a c +b \right ) x^{3} c^{2} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\) | \(571\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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