Optimal. Leaf size=639 \[ -\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {d} (b c-a d)^2 (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {e^{3/2} \sqrt {f} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 (b c-a d)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 e^{3/2} (b e-a f) \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.51, antiderivative size = 639, 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 = {558, 555, 553,
422, 540, 541, 539, 429} \begin {gather*} \frac {b^2 e^{3/2} \sqrt {c+d x^2} (b e-a f) \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 {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-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) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )-3 b c \left (c^2 f^2-6 c d e f+6 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 {d} \sqrt {c+d x^2} (b c-a d)^2 (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {b \sqrt {d} \sqrt {e+f x^2} (b e-a f) 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 {x \sqrt {e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {x \sqrt {e+f x^2} (d e-c f)}{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 558
Rubi steps
\begin {align*} \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx &=-\frac {\int \frac {\sqrt {e+f x^2} \left (2 b c d e-a d^2 e-b c^2 f+d^2 (b e-a f) x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx}{(b c-a d)^2}+\frac {(b (b e-a f)) \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 e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {\int \frac {-d e (b c (9 d e-4 c f)-a d (4 d e+c f))-d f (b c (8 d e-3 c f)-a d (3 d e+2 c f)) x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c d (b c-a d)^2}+\frac {\left (b^2 (b e-a f)\right ) \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 {(b d (b e-a f)) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^3}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \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^2 e^{3/2} (b e-a f) \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 (d e-c f) (9 b c (2 d e-c f)-a d (8 d e+c f))+d f (d e-c f) (3 b c (3 d e-c f)-2 a d (2 d e+c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 d (b c-a d)^2 (d e-c f)}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \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^2 e^{3/2} (b e-a f) \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 {\left (e f \left (a d (4 d e-c f)-b \left (9 c d e-6 c^2 f\right )\right )\right ) \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)}+\frac {\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\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)}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \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 {\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+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} \sqrt {d} (b c-a d)^2 (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} \left (a d (4 d e-c f)-b \left (9 c d e-6 c^2 f\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 )}{15 c^3 (b c-a d)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 e^{3/2} (b e-a f) \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 = 7.90, size = 570, normalized size = 0.89 \begin {gather*} \frac {-a \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) (3 b c (-3 d e+c f)+2 a d (2 d e+c f)) \left (c+d x^2\right )+\left (a^2 d^2 \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )+3 b^2 c^2 \left (11 d^2 e^2-11 c d e f+c^2 f^2\right )+2 a b c d \left (-13 d^2 e^2+3 c d e f+7 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 e \left (-3 b^2 c^2 \left (11 d^2 e^2-11 c d e f+c^2 f^2\right )+a^2 d^2 \left (-8 d^2 e^2+3 c d e f+2 c^2 f^2\right )-2 a b c d \left (-13 d^2 e^2+3 c d e f+7 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 (3 b^2 c^2 e (11 d e-8 c f)+a^2 d^2 e (8 d e+c f)+a b c \left (-26 d^2 e^2-7 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 c^3 (b e-a f)^2 \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) \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. \(6210\) vs.
\(2(723)=1446\).
time = 0.16, size = 6211, normalized size = 9.72
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3112\) |
default | \(\text {Expression too large to display}\) | \(6211\) |
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(-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 \frac {{\left (f\,x^2+e\right )}^{3/2}}{\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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