Optimal. Leaf size=544 \[ -\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\sqrt {e} \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.45, antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {e^{3/2} \sqrt {c+d x^2} \left (7 a d f (d e-9 c f)-b \left (-3 c^2 f^2-9 c d e f+4 d^2 e^2\right )\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{5/2} \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} \sqrt {e+f x^2} \left (7 a d f (3 c f+d e)-b \left (6 c^2 f^2-6 c d e f+4 d^2 e^2\right )\right )}{105 d f^2}-\frac {x \sqrt {c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right )}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 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 ) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx &=\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (-(b c-7 a d) e+(b d e-2 b c f+7 a d f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{7 d}\\ &=\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\int \frac {\sqrt {c+d x^2} \left (-c e (b d e+3 b c f-28 a d f)+\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x^2\right )}{\sqrt {e+f x^2}} \, dx}{35 d f}\\ &=\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\int \frac {-c e \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right )+\left (-7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d f^2}\\ &=\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}-\frac {\left (c e \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d f^2}-\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d f^2}\\ &=-\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}-\frac {e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d f^{5/2} \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 (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{105 d^2 f^2}\\ &=-\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\sqrt {e} \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\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 )}{105 d^2 f^{5/2} \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} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d f^{5/2} \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 = 3.37, size = 373, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (6 c f+d \left (e+3 f x^2\right )\right )+b \left (3 c^2 f^2+3 c d f \left (3 e+8 f x^2\right )+d^2 \left (-4 e^2+3 e f x^2+15 f^2 x^4\right )\right )\right )+i e \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+b \left (-8 d^3 e^3+19 c d^2 e^2 f-9 c^2 d e f^2+6 c^3 f^3\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) \left (-14 a d f (d e-3 c f)+b \left (8 d^2 e^2-15 c d e f+3 c^2 f^2\right )\right ) \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 )}{105 d \sqrt {\frac {d}{c}} f^3 \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. \(1331\) vs.
\(2(566)=1132\).
time = 0.18, size = 1332, normalized size = 2.45
method | result | size |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b d \,x^{5} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{7}+\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d f}+\frac {\left (2 a c d f +a \,d^{2} e +b \,c^{2} f +\frac {9 b c d e}{7}-\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (c^{2} a e -\frac {\left (2 a c d f +a \,d^{2} e +b \,c^{2} f +\frac {9 b c d e}{7}-\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (c^{2} a f +2 a c d e +b \,c^{2} e -\frac {3 \left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) c e}{5 d f}-\frac {\left (2 a c d f +a \,d^{2} e +b \,c^{2} f +\frac {9 b c d e}{7}-\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(695\) |
risch | \(\frac {x \left (15 b \,x^{4} d^{2} f^{2}+21 a \,d^{2} f^{2} x^{2}+24 b c d \,f^{2} x^{2}+3 b \,d^{2} e f \,x^{2}+42 a c d \,f^{2}+7 a \,d^{2} e f +3 b \,c^{2} f^{2}+9 b c d e f -4 b \,d^{2} e^{2}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{105 d \,f^{2}}+\frac {\left (-\frac {\left (21 a \,c^{2} d \,f^{3}+49 a c \,d^{2} e \,f^{2}-14 a \,d^{3} e^{2} f -6 b \,c^{3} f^{3}+9 b \,c^{2} d e \,f^{2}-19 b c \,d^{2} e^{2} f +8 b \,d^{3} e^{3}\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 {63 a \,c^{2} d e \,f^{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}}-\frac {7 a c \,d^{2} e^{2} 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 {3 b \,c^{3} e \,f^{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}}-\frac {9 b \,c^{2} d \,e^{2} 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 {4 b c \,d^{2} e^{3} \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 )}}{105 f^{2} d \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(834\) |
default | \(\text {Expression too large to display}\) | \(1332\) |
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 ) \left (c + d x^{2}\right )^{\frac {3}{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 )\,{\left (d\,x^2+c\right )}^{3/2}\,\sqrt {f\,x^2+e} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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