Optimal. Leaf size=630 \[ -\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {d} \left (a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b c \left (9 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.48, antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {560, 555, 553,
422, 540, 541, 539, 429} \begin {gather*} \frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c \sqrt {f} \sqrt {e+f x^2} (b c-a d)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} \sqrt {c+d x^2} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 \sqrt {e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b c \left (4 c^2 f^2-14 c d e f+9 d^2 e^2\right )\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {d x \sqrt {e+f x^2} (b c (9 d e-8 c f)-a d (4 d e-3 c f))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2 (d e-c f)}-\frac {d x \sqrt {e+f x^2}}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 539
Rule 540
Rule 541
Rule 553
Rule 555
Rule 560
Rubi steps
\begin {align*} \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx &=\frac {b^2 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^2}-\frac {d \int \frac {\left (2 b c-a d+b d x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx}{(b c-a d)^2}\\ &=-\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {b^3 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^3}-\frac {\left (b^2 d\right ) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^3}+\frac {\int \frac {-d (9 b c-4 a d) e-d (8 b c-3 a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c (b c-a d)^2}\\ &=-\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\int \frac {d e (b c (18 d e-19 c f)-a d (8 d e-9 c f))+d f (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)}\\ &=-\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(d e f (b c (9 d e-11 c f)-2 a d (2 d e-3 c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)^2}+\frac {\left (d \left (a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b c \left (9 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)^2}\\ &=-\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {d} \left (a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b c \left (9 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} (b c (9 d e-11 c f)-2 a d (2 d e-3 c 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 c^3 (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \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 = 6.06, size = 584, normalized size = 0.93 \begin {gather*} \frac {-a d \sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)^2+c (b c-a d) (-d e+c f) (a d (4 d e-3 c f)+b c (-9 d e+8 c f)) \left (c+d x^2\right )+\left (a b c d \left (-26 d^2 e^2+41 c d e f-11 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (33 d^2 e^2-58 c d e f+23 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )-i \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (a d e \left (a b c d \left (-26 d^2 e^2+41 c d e f-11 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (33 d^2 e^2-58 c d e f+23 c^2 f^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-(d e-c f) \left (-a \left (2 a b c d^2 e (13 d e-14 c f)+a^2 d^3 e (-8 d e+9 c f)+b^2 c^2 \left (-33 d^2 e^2+49 c d e f-15 c^2 f^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+15 b^2 c^3 (b e-a f) (-d e+c f) \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )\right )}{15 a c^3 \sqrt {\frac {d}{c}} (b c-a d)^3 (d e-c f)^2 \left (c+d x^2\right )^{5/2} \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. \(6244\) vs.
\(2(714)=1428\).
time = 0.18, size = 6245, normalized size = 9.91
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3138\) |
default | \(\text {Expression too large to display}\) | \(6245\) |
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(-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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e + f x^{2}}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {7}{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 \frac {\sqrt {f\,x^2+e}}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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