Optimal. Leaf size=373 \[ -\frac {f x \sqrt {c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f))}{3 c^2 d^3 \sqrt {e+f x^2}}+\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f)) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 d^3 \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (4 b c-a d) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 d^2 \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {526, 531, 418, 492, 411} \[ \frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (4 b c-a d) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 d^2 \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {f x \sqrt {c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f))}{3 c^2 d^3 \sqrt {e+f x^2}}+\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f)) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 d^3 \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 492
Rule 526
Rule 531
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx &=-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {\sqrt {e+f x^2} \left (-(b c+2 a d) e-(4 b c-a d) f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d}\\ &=\frac {(b c (d e-4 c f)+a d (2 d e+c f)) x \sqrt {e+f x^2}}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {c (4 b c-a d) e f-f (b c (d e-8 c f)+2 a d (d e+c f)) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c^2 d^2}\\ &=\frac {(b c (d e-4 c f)+a d (2 d e+c f)) x \sqrt {e+f x^2}}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {((4 b c-a d) e f) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c d^2}-\frac {(f (b c (d e-8 c f)+2 a d (d e+c f))) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c^2 d^2}\\ &=-\frac {f (b c (d e-8 c f)+2 a d (d e+c f)) x \sqrt {c+d x^2}}{3 c^2 d^3 \sqrt {e+f x^2}}+\frac {(b c (d e-4 c f)+a d (2 d e+c f)) x \sqrt {e+f x^2}}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {(4 b c-a d) e^{3/2} \sqrt {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 )}{3 c^2 d^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(e f (b c (d e-8 c f)+2 a d (d e+c f))) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 d^3}\\ &=-\frac {f (b c (d e-8 c f)+2 a d (d e+c f)) x \sqrt {c+d x^2}}{3 c^2 d^3 \sqrt {e+f x^2}}+\frac {(b c (d e-4 c f)+a d (2 d e+c f)) x \sqrt {e+f x^2}}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \sqrt {f} (b c (d e-8 c f)+2 a d (d e+c 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 c^2 d^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(4 b c-a d) e^{3/2} \sqrt {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 )}{3 c^2 d^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] time = 1.23, size = 296, normalized size = 0.79 \[ \frac {\left (\frac {d}{c}\right )^{3/2} \left (x \sqrt {\frac {d}{c}} \left (e+f x^2\right ) \left (a d \left (c^2 f+c d \left (3 e+2 f x^2\right )+2 d^2 e x^2\right )+b c \left (-4 c^2 f-5 c d f x^2+d^2 e x^2\right )\right )+i e \left (c+d x^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (b c (4 c f-d e)-a d (c f+2 d e)) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i e \left (c+d x^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (b c (8 c f-d e)-2 a d (c f+d e)) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{3 d^4 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b f x^{4} + {\left (b e + a f\right )} x^{2} + a e\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1231, normalized size = 3.30 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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