Optimal. Leaf size=385 \[ -\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} d^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} (2 a d (2 d e-3 c f)+b c (d e+c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 d (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {540, 541, 539,
429, 422} \begin {gather*} -\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 d \sqrt {e+f x^2} (d e-c f)^2 \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 (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+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} d^{3/2} \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {x \sqrt {e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 539
Rule 540
Rule 541
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac {\int \frac {-(b c+4 a d) e-(2 b c+3 a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c d}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {e (a d (8 d e-9 c f)+b c (2 d e-c f))+f (d (b c+4 a d) e-c (2 b c+3 a d) f) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 d (d e-c f)}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(e f (2 a d (2 d e-3 c f)+b c (d e+c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 d (d e-c f)^2}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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 d (d e-c f)^2}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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} d^{3/2} (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 {e^{3/2} \sqrt {f} (2 a d (2 d e-3 c f)+b c (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 d (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}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.14, size = 379, normalized size = 0.98 \begin {gather*} \frac {-\sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (3 c^2 (b c-a d) (d e-c f)^2-c (d e-c f) (a d (4 d e-3 c f)+b c (d e-2 c f)) \left (c+d x^2\right )-\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )+i e \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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) (b c (-2 d e+c f)+a d (-8 d e+9 c f)) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{15 c^4 \left (\frac {d}{c}\right )^{3/2} (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. \(2860\) vs.
\(2(419)=838\).
time = 0.15, size = 2861, normalized size = 7.43
method | result | size |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (a d -b c \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d^{4} c \left (x^{2}+\frac {c}{d}\right )^{3}}+\frac {\left (3 a c d f -4 a \,d^{2} e +2 b \,c^{2} f -b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{15 d^{3} c^{2} \left (c f -d e \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) x \left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d^{2} c^{3} \left (c f -d e \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {f \left (3 a c d f -4 a \,d^{2} e +2 b \,c^{2} f -b c d e \right )}{15 c^{2} \left (c f -d e \right ) d^{2}}-\frac {3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}}{15 d^{2} \left (c f -d e \right ) c^{3}}-\frac {e \left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d \,c^{3} \left (c f -d e \right )^{2}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{15 d \,c^{3} \left (c f -d e \right )^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(756\) |
default | \(\text {Expression too large to display}\) | \(2861\) |
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 \frac {\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}{\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 {\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}}{{\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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