Optimal. Leaf size=298 \[ -\frac {3 d e^2 x \sqrt {a+c x^4}}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {3 d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}+\frac {d \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt {a+c x^4}} \]
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Rubi [A]
time = 0.09, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1868, 1212,
226, 1210} \begin {gather*} \frac {d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt {a+c x^4}}+\frac {3 d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}-\frac {3 d e^2 x \sqrt {a+c x^4}}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 1210
Rule 1212
Rule 1868
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}-\frac {\int \frac {-d^3+3 d e^2 x^2}{\sqrt {a+c x^4}} \, dx}{2 a}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {\left (3 d e^2\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} \sqrt {c}}+\frac {\left (d \left (d^2-\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 a}\\ &=-\frac {3 d e^2 x \sqrt {a+c x^4}}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {3 d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}+\frac {d \left (d^2-\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.08, size = 126, normalized size = 0.42 \begin {gather*} \frac {-a e^3+c d^3 x+3 c d^2 e x^2+c d^3 x \sqrt {1+\frac {c x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^4}{a}\right )+2 c d e^2 x^3 \sqrt {1+\frac {c x^4}{a}} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {c x^4}{a}\right )}{2 a c \sqrt {a+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.21, size = 261, normalized size = 0.88
method | result | size |
elliptic | \(-\frac {2 c \left (-\frac {3 d \,e^{2} x^{3}}{4 a c}-\frac {3 d^{2} e \,x^{2}}{4 c a}-\frac {d^{3} x}{4 a c}+\frac {e^{3}}{4 c^{2}}\right )}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 i d \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(245\) |
default | \(-\frac {e^{3}}{2 c \sqrt {c \,x^{4}+a}}+3 d \,e^{2} \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+\frac {3 d^{2} e \,x^{2}}{2 \sqrt {c \,x^{4}+a}\, a}+d^{3} \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\) | \(261\) |
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 (d + e x\right )^{3}}{\left (a + c x^{4}\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 \frac {{\left (d+e\,x\right )}^3}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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