Optimal. Leaf size=608 \[ \frac {d \left (7 c e-\frac {2 d e^2}{f}+\frac {3 c^2 f}{d}\right ) x \sqrt {c+d x^2}}{15 b \sqrt {e+f x^2}}+\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {\sqrt {e} \left (15 a^2 d^2 f^2-5 a b d f (d e+7 c f)+b^2 \left (-2 d^2 e^2+12 c d e f+23 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d e^{3/2} \left (-40 a b c d f+15 a^2 d^2 f+b^2 c (-d e+34 c f)\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 c f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.52, antiderivative size = 776, normalized size of antiderivative = 1.28, number of steps
used = 14, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {559, 427, 542,
545, 429, 506, 422, 557, 553} \begin {gather*} \frac {d e^{3/2} \sqrt {c+d x^2} (5 b c-3 a d) (b c-a d) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \sqrt {f} \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} (b c-a d)^3 \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 b^3 c \sqrt {f} \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} (b c-a d) (-3 a d f+4 b c f+b d e) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {f} \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} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt {e+f x^2}}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)}{3 b^2}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b 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 {d e^{3/2} \sqrt {c+d x^2} (d e-9 c f) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b 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 {d x \sqrt {c+d x^2} \left (\frac {3 c^2 f}{d}+7 c e-\frac {2 d e^2}{f}\right )}{15 b \sqrt {e+f x^2}}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {2 d x \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-3 c f)}{15 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 427
Rule 429
Rule 506
Rule 542
Rule 545
Rule 553
Rule 557
Rule 559
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{a+b x^2} \, dx &=\frac {d \int \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx}{b}+\frac {(b c-a d) \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{a+b x^2} \, dx}{b}\\ &=\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {(d (b c-a d)) \int \frac {\left (2 b c-a d+b d x^2\right ) \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx}{b^3}+\frac {(b c-a d)^3 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}+\frac {d \int \frac {\sqrt {e+f x^2} \left (-c (d e-5 c f)-2 d (d e-3 c f) x^2\right )}{\sqrt {c+d x^2}} \, dx}{5 b f}\\ &=\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {(b c-a d)^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 b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d) \int \frac {d (5 b c-3 a d) e+d (b d e+4 b c f-3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}+\frac {\int \frac {-c d e (d e-9 c f)-d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f}\\ &=\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {(b c-a d)^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 b^3 c \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 (5 b c-3 a d) (b c-a d) e) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}-\frac {(c d e (d e-9 c f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f}+\frac {(d (b c-a d) (b d e+4 b c f-3 a d f)) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}-\frac {\left (d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f}\\ &=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {e+f x^2}}-\frac {\left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b f \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \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^{3/2} (d e-9 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 b 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 {(b c-a d)^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 b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {((b c-a d) e (b d e+4 b c f-3 a d f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b^3}+\frac {\left (e \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 b f}\\ &=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {e+f x^2}}-\frac {\left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b f \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {(b c-a d) \sqrt {e} (b d e+4 b c f-3 a d f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^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 {\sqrt {e} \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\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 b 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 {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \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^{3/2} (d e-9 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 b 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 {(b c-a d)^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 b^3 c \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 = 5.62, size = 456, normalized size = 0.75 \begin {gather*} \frac {-i a b d e \left (15 a^2 d^2 f^2-5 a b d f (d e+7 c f)+b^2 \left (-2 d^2 e^2+12 c d e f+23 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 a \left (45 a^2 b c d^2 f^3-15 a^3 d^3 f^3+5 a b^2 d f \left (d^2 e^2-c d e f-9 c^2 f^2\right )+b^3 \left (2 d^3 e^3-13 c d^2 e^2 f+11 c^2 d e f^2+15 c^3 f^3\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 )+f \left (a b^2 d \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (11 b c f-5 a d f+b d \left (e+3 f x^2\right )\right )-15 i (b c-a d)^3 f (b e-a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{15 a b^4 \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. \(1890\) vs.
\(2(649)=1298\).
time = 0.23, size = 1891, normalized size = 3.11
method | result | size |
risch | \(-\frac {d x \left (-3 b d \,x^{2} f +5 a d f -11 b c f -b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 f \,b^{2}}+\frac {\left (-\frac {d \left (15 a^{2} d^{2} f^{2}-35 a b c d \,f^{2}-5 a b \,d^{2} e f +23 b^{2} c^{2} f^{2}+12 b^{2} c d e f -2 b^{2} d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}-\frac {\left (15 a^{3} d^{3} f^{2}-45 a^{2} b c \,d^{2} f^{2}-15 a^{2} b \,d^{3} e f +45 a \,b^{2} c^{2} d \,f^{2}+40 a \,b^{2} c \,d^{2} e f -15 b^{3} c^{3} f^{2}-34 b^{3} c^{2} d e f +b^{3} c \,d^{2} e^{2}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {15 \left (a^{4} d^{3} f -3 a^{3} b c \,d^{2} f -a^{3} b \,d^{3} e +3 a^{2} b^{2} c^{2} d f +3 a^{2} b^{2} c \,d^{2} e -a \,b^{3} c^{3} f -3 a \,b^{3} c^{2} d e +c^{3} e \,b^{4}\right ) f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{b^{2} a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 f \,b^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(657\) |
default | \(\text {Expression too large to display}\) | \(1891\) |
elliptic | \(\text {Expression too large to display}\) | \(2436\) |
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 {\left (c + d x^{2}\right )^{\frac {5}{2}} \sqrt {e + f x^{2}}}{a + b 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 \frac {{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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