Optimal. Leaf size=608 \[ -\frac {\sqrt {e} \sqrt {c+d x^2} \left (15 a^2 d^2 f^2-5 a b d f (7 c f+d e)+b^2 \left (23 c^2 f^2+12 c d e f-2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {d e^{3/2} \sqrt {c+d x^2} \left (15 a^2 d^2 f-40 a b c d f+b^2 c (34 c f-d e)\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 c f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {e^{3/2} \sqrt {c+d x^2} (b c-a d)^3 \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt {e+f x^2}}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)}{3 b^2}+\frac {d x \sqrt {c+d x^2} \left (\frac {3 c^2 f}{d}+7 c e-\frac {2 d e^2}{f}\right )}{15 b \sqrt {e+f x^2}}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {2 d x \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-3 c f)}{15 b f} \]
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Rubi [A] time = 0.76, 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 = {545, 416, 528, 531, 418, 492, 411, 543, 539} \[ \frac {d e^{3/2} \sqrt {c+d x^2} (5 b c-3 a d) (b c-a d) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {e^{3/2} \sqrt {c+d x^2} (b c-a d)^3 \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)}{3 b^2}+\frac {x \sqrt {c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {d x \sqrt {c+d x^2} \left (\frac {3 c^2 f}{d}+7 c e-\frac {2 d e^2}{f}\right )}{15 b \sqrt {e+f x^2}}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {d e^{3/2} \sqrt {c+d x^2} (d e-9 c f) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {2 d x \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-3 c f)}{15 b f} \]
Antiderivative was successfully verified.
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Rule 411
Rule 416
Rule 418
Rule 492
Rule 528
Rule 531
Rule 539
Rule 543
Rule 545
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] time = 2.80, size = 456, normalized size = 0.75 \[ \frac {-i a b d e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (15 a^2 d^2 f^2-5 a b d f (7 c f+d e)+b^2 \left (23 c^2 f^2+12 c d e f-2 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (-15 a^3 d^3 f^3+45 a^2 b c d^2 f^3+5 a b^2 d f \left (-9 c^2 f^2-c d e f+d^2 e^2\right )+b^3 \left (15 c^3 f^3+11 c^2 d e f^2-13 c d^2 e^2 f+2 d^3 e^3\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+f \left (a b^2 d x \sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-5 a d f+11 b c f+b d \left (e+3 f x^2\right )\right )-15 i f \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (b c-a d)^3 (b e-a f) \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{15 a b^4 f^2 \sqrt {\frac {d}{c}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
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 \[ \int \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1891, normalized size = 3.11 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}}{b\,x^2+a} \,d x \]
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 \[ \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}} \sqrt {e + f x^{2}}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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