Optimal. Leaf size=357 \[ \frac {\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 b^2 d^2}-\frac {(4 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 b d^2}+\frac {x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d}-\frac {\sqrt {c} \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} (4 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.26, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1986, 489, 596,
545, 429, 506, 422} \begin {gather*} -\frac {\sqrt {c} \left (-2 a^2 d^2-3 a b c d+8 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 )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 b^2 d^2}+\frac {c^{3/2} (4 b c-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 )}{15 b d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \left (c+d x^2\right ) (4 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 b d^2}+\frac {x^3 \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 489
Rule 506
Rule 545
Rule 596
Rule 1986
Rubi steps
\begin {align*} \int x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx &=\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx}{\sqrt {a+b x^2}}\\ &=\frac {x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d}-\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2 \left (3 a c+(4 b c-a d) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 d \sqrt {a+b x^2}}\\ &=-\frac {(4 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 b d^2}+\frac {x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d}+\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a c (4 b c-a d)+\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b d^2 \sqrt {a+b x^2}}\\ &=-\frac {(4 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 b d^2}+\frac {x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d}+\frac {\left (a c (4 b c-a d) \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}{15 b d^2 \sqrt {a+b x^2}}+\frac {\left (\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \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}{15 b d^2 \sqrt {a+b x^2}}\\ &=\frac {\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 b^2 d^2}-\frac {(4 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 b d^2}+\frac {x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d}+\frac {c^{3/2} (4 b c-a d) \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 )}{15 b d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\left (c \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \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}{15 b^2 d^2 \sqrt {a+b x^2}}\\ &=\frac {\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 b^2 d^2}-\frac {(4 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 b d^2}+\frac {x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d}-\frac {\sqrt {c} \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \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 )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} (4 b c-a d) \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 )}{15 b d^{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 = 1.58, size = 255, normalized size = 0.71 \begin {gather*} \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-4 b c+a d+3 b d x^2\right )+i c \left (-8 b^2 c^2+3 a b c d+2 a^2 d^2\right ) \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 c \left (-8 b^2 c^2+7 a b c d+a^2 d^2\right ) \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 )}{15 b \sqrt {\frac {b}{a}} d^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 552, normalized size = 1.55
method | result | size |
risch | \(\frac {x \left (3 b d \,x^{2}+a d -4 b c \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{15 d^{2} b}-\frac {\left (-\frac {2 \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}+\frac {a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {4 a b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{15 b \,d^{2} \left (b \,x^{2}+a \right )}\) | \(490\) |
default | \(\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+4 \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}-\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+\sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, d^{3} b \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(552\) |
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 x^4\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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