Optimal. Leaf size=381 \[ \frac {\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^2 f \sqrt {e+f x^2}}+\frac {(b d e-2 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\sqrt {e} \left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^2 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {e^{3/2} (b d e+b c f-10 a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {542, 545, 429,
506, 422} \begin {gather*} -\frac {\sqrt {e} \sqrt {c+d x^2} \left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^2 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {e^{3/2} \sqrt {c+d x^2} (-10 a d f+b c f+b d e) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} \left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right )}{15 d^2 f \sqrt {e+f x^2}}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-2 b c f+b d e)}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 506
Rule 542
Rule 545
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx &=\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}+\frac {\int \frac {\sqrt {c+d x^2} \left (-(b c-5 a d) e+(b d e-2 b c f+5 a d f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{5 d}\\ &=\frac {(b d e-2 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}+\frac {\int \frac {-c e (b d e+b c f-10 a d f)+\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d f}\\ &=\frac {(b d e-2 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {(c e (b d e+b c f-10 a d f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d f}+\frac {\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d f}\\ &=\frac {\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^2 f \sqrt {e+f x^2}}+\frac {(b d e-2 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {e^{3/2} (b d e+b c f-10 a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (e \left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 d^2 f}\\ &=\frac {\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^2 f \sqrt {e+f x^2}}+\frac {(b d e-2 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\sqrt {e} \left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^2 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {e^{3/2} (b d e+b c f-10 a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.70, size = 267, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (b c f+5 a d f+b d \left (e+3 f x^2\right )\right )+i e \left (-5 a d f (d e+c f)+2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i e (-d e+c f) (-2 b d e+b c f+5 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{15 d \sqrt {\frac {d}{c}} f^2 \sqrt {c+d x^2} \sqrt {e+f 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. \(864\) vs.
\(2(409)=818\).
time = 0.13, size = 865, normalized size = 2.27
method | result | size |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b \,x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5}+\frac {\left (a d f +b c f +b d e -\frac {b \left (4 c f +4 d e \right )}{5}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (a c e -\frac {\left (a d f +b c f +b d e -\frac {b \left (4 c f +4 d e \right )}{5}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (a c f +a d e +\frac {2 b c e}{5}-\frac {\left (a d f +b c f +b d e -\frac {b \left (4 c f +4 d e \right )}{5}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(431\) |
risch | \(\frac {x \left (3 b d \,x^{2} f +5 a d f +b c f +b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 d f}+\frac {\left (-\frac {\left (5 a c d \,f^{2}+5 a \,d^{2} e f -2 b \,c^{2} f^{2}+2 b c d e f -2 b \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}+\frac {10 a c d e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \,c^{2} e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b c d \,e^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 d f \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(538\) |
default | \(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, \left (3 \sqrt {-\frac {d}{c}}\, b \,d^{2} f^{3} x^{7}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} f^{3} x^{5}+4 \sqrt {-\frac {d}{c}}\, b c d \,f^{3} x^{5}+4 \sqrt {-\frac {d}{c}}\, b \,d^{2} e \,f^{2} x^{5}+5 \sqrt {-\frac {d}{c}}\, a c d \,f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} e \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, b \,c^{2} f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, b c d e \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, b \,d^{2} e^{2} f \,x^{3}+5 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e \,f^{2}-5 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2} f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e \,f^{2}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2} f +2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,d^{2} e^{3}+5 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e \,f^{2}+5 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2} f -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e \,f^{2}+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2} f -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,d^{2} e^{3}+5 \sqrt {-\frac {d}{c}}\, a c d e \,f^{2} x +\sqrt {-\frac {d}{c}}\, b \,c^{2} e \,f^{2} x +\sqrt {-\frac {d}{c}}\, b c d \,e^{2} f x \right )}{15 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) d \,f^{2} \sqrt {-\frac {d}{c}}}\) | \(865\) |
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 \left (a + b x^{2}\right ) \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}\, 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 \left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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