Optimal. Leaf size=405 \[ \frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.42, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1985, 1986,
478, 595, 596, 545, 429, 506, 422} \begin {gather*} -\frac {\sqrt {c} \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 a d^2}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (7 b-a c) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d}-\frac {x^3 \left (a c+a d x^2+b\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 478
Rule 506
Rule 545
Rule 595
Rule 596
Rule 1985
Rule 1986
Rubi steps
\begin {align*} \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^4 \left (b+a \left (c+d x^2\right )\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^4 \left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2 \sqrt {b+a c+a d x^2} \left (3 (b+a c)+6 a d x^2\right )}{\sqrt {c+d x^2}} \, dx}{d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2 \left (3 (5 b-a c) (b+a c) d+3 a (7 b-a c) d^2 x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {3 a c (7 b-a c) (b+a c) d^2-3 a \left (b^2-14 a b c+a^2 c^2\right ) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d^4 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c (7 b-a c) (b+a c) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\left (b^2-14 a b c+a^2 c^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} (7 b-a c) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 a d^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\sqrt {c} \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} (7 b-a c) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.59, size = 308, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} x \left (-a^2 \left (c-d x^2\right ) \left (c+d x^2\right )^2+b^2 \left (7 c+2 d x^2\right )+3 a b \left (2 c^2+3 c d x^2+d^2 x^4\right )\right )-i c \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )+8 i b c (b-a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{5 d^2 \sqrt {\frac {a d}{b+a c}} \left (b+a \left (c+d x^2\right )\right )} \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. \(1097\) vs.
\(2(441)=882\).
time = 0.10, size = 1098, normalized size = 2.71 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 \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 x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{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 x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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