Optimal. Leaf size=480 \[ -\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac {\sqrt {d} \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.46, antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1986, 479, 597,
545, 429, 506, 422} \begin {gather*} -\frac {\sqrt {d} e \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {e \left (c+d x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4 x}+\frac {d e x \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}-\frac {b \sqrt {d} e (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 c^3 x^3}+\frac {e \left (c+d x^2\right ) (5 b c-6 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 c^2 d x^5}-\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 479
Rule 506
Rule 545
Rule 597
Rule 1986
Rubi steps
\begin {align*} \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx &=\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}-\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a (5 b c-6 a d)+b (4 b c-5 a d) x^2}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{c d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}+\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {3 a^2 d (7 b c-8 a d)+3 a b d (5 b c-6 a d) x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 a c^2 d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {-3 a^2 d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right )+3 a^2 b d^2 (7 b c-8 a d) x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^2 c^3 d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}+\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {-3 a^3 b c d^2 (7 b c-8 a d)+3 a^2 b d^2 \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^3 c^4 d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac {\left (b d (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 c^3 \sqrt {a+b x^2}}+\frac {\left (b d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 a c^4 \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac {b \sqrt {d} (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\left (d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 a c^3 \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac {\sqrt {d} \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.34, size = 357, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {\frac {b}{a}} e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} \left (b^3 c^2 x^6 \left (c+d x^2\right )+a b^2 c x^4 \left (3 c^2-8 c d x^2-16 d^2 x^4\right )+a^2 b x^2 \left (3 c^3-11 c^2 d x^2-8 c d^2 x^4+16 d^3 x^6\right )+a^3 \left (c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right )\right )+i b c \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c \left (b^2 c^2-9 a b c d+8 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{5 b c^4 x^5 \left (a+b x^2\right )} \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. \(1196\) vs.
\(2(512)=1024\).
time = 0.09, size = 1197, normalized size = 2.49 Too large to display
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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