Optimal. Leaf size=630 \[ \frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c \sqrt {f} \sqrt {e+f x^2} (b c-a d)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} \sqrt {c+d x^2} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 \sqrt {e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {d x \sqrt {e+f x^2} (b c (9 d e-8 c f)-a d (4 d e-3 c f))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2 (d e-c f)}+\frac {\sqrt {d} \sqrt {e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b c \left (4 c^2 f^2-14 c d e f+9 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {d x \sqrt {e+f x^2}}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)} \]
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Rubi [A] time = 0.72, 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 = {546, 541, 539, 411, 526, 527, 525, 418} \[ \frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c \sqrt {f} \sqrt {e+f x^2} (b c-a d)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b c \left (4 c^2 f^2-14 c d e f+9 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} \sqrt {c+d x^2} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 \sqrt {e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {d x \sqrt {e+f x^2} (b c (9 d e-8 c f)-a d (4 d e-3 c f))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2 (d e-c f)}-\frac {d x \sqrt {e+f x^2}}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 525
Rule 526
Rule 527
Rule 539
Rule 541
Rule 546
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] time = 3.08, size = 584, normalized size = 0.93 \[ \frac {-a d x \sqrt {\frac {d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a^2 d^2 \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+a b c d \left (-11 c^2 f^2+41 c d e f-26 d^2 e^2\right )+b^2 c^2 \left (23 c^2 f^2-58 c d e f+33 d^2 e^2\right )\right )+3 c^2 (b c-a d)^2 (d e-c f)^2+c \left (c+d x^2\right ) (b c-a d) (c f-d e) (a d (4 d e-3 c f)+b c (8 c f-9 d e))\right )-i \sqrt {\frac {d x^2}{c}+1} \left (c+d x^2\right )^2 \sqrt {\frac {f x^2}{e}+1} \left (a d e \left (a^2 d^2 \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+a b c d \left (-11 c^2 f^2+41 c d e f-26 d^2 e^2\right )+b^2 c^2 \left (23 c^2 f^2-58 c d e f+33 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-(d e-c f) \left (15 b^2 c^3 (b e-a f) (c f-d e) \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-a \left (a^2 d^3 e (9 c f-8 d e)+2 a b c d^2 e (13 d e-14 c f)+b^2 c^2 \left (-15 c^2 f^2+49 c d e f-33 d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )\right )}{15 a c^3 \sqrt {\frac {d}{c}} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2} (b c-a d)^3 (d e-c f)^2} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 6245, normalized size = 9.91 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {f\,x^2+e}}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\sqrt {e + f x^{2}}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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