3.200 \(\int \frac{1}{(d+e x) \left (a+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=827 \[ -\frac{\tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{c x^4+a}}\right ) e^5}{2 \left (c d^4+a e^4\right )^{3/2}}+\frac{\sqrt [4]{c} d \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) e^4}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) e^4}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{c d^4+a e^4}{d^2 e^2}} x}{\sqrt{c x^4+a}}\right ) e^2}{2 d^3 \left (-\frac{c d^4+a e^4}{d^2 e^2}\right )^{3/2}}+\frac{\sqrt [4]{c} d \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) e^2}{2 a^{3/4} \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}-\frac{\sqrt{c} d x \sqrt{c x^4+a} e^2}{2 a \left (c d^4+a e^4\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{\left (a e^2-c d^2 x^2\right ) e}{2 a \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}+\frac{c d x \left (d^2+e^2 x^2\right )}{2 a \left (c d^4+a e^4\right ) \sqrt{c x^4+a}} \]

[Out]

(e*(a*e^2 - c*d^2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c*d*x*(d^2 + e^
2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (Sqrt[c]*d*e^2*x*Sqrt[a + c*x^4]
)/(2*a*(c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)) - (e^2*ArcTan[(Sqrt[-((c*d^4 + a
*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*d^3*(-((c*d^4 + a*e^4)/(d^2*e^2)))^(3/
2)) - (e^5*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(
2*(c*d^4 + a*e^4)^(3/2)) + (c^(1/4)*d*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^
4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*
a^(3/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c^(1/4)*d*(Sqrt[c]*d^2 - Sqrt[a]*e^2
)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[
2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4])
+ (c^(1/4)*d*e^4*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^
2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d^2 + S
qrt[a]*e^2)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*
(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(
Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a
^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)*
Sqrt[a + c*x^4])

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Rubi [A]  time = 1.72462, antiderivative size = 827, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632 \[ -\frac{\tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{c x^4+a}}\right ) e^5}{2 \left (c d^4+a e^4\right )^{3/2}}+\frac{\sqrt [4]{c} d \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) e^4}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) e^4}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{c d^4+a e^4}{d^2 e^2}} x}{\sqrt{c x^4+a}}\right ) e^2}{2 d^3 \left (-\frac{c d^4+a e^4}{d^2 e^2}\right )^{3/2}}+\frac{\sqrt [4]{c} d \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) e^2}{2 a^{3/4} \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}-\frac{\sqrt{c} d x \sqrt{c x^4+a} e^2}{2 a \left (c d^4+a e^4\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{\left (a e^2-c d^2 x^2\right ) e}{2 a \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \left (c d^4+a e^4\right ) \sqrt{c x^4+a}}+\frac{c d x \left (d^2+e^2 x^2\right )}{2 a \left (c d^4+a e^4\right ) \sqrt{c x^4+a}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((d + e*x)*(a + c*x^4)^(3/2)),x]

[Out]

(e*(a*e^2 - c*d^2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c*d*x*(d^2 + e^
2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (Sqrt[c]*d*e^2*x*Sqrt[a + c*x^4]
)/(2*a*(c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)) - (e^2*ArcTan[(Sqrt[-((c*d^4 + a
*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*d^3*(-((c*d^4 + a*e^4)/(d^2*e^2)))^(3/
2)) - (e^5*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(
2*(c*d^4 + a*e^4)^(3/2)) + (c^(1/4)*d*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^
4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*
a^(3/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c^(1/4)*d*(Sqrt[c]*d^2 - Sqrt[a]*e^2
)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[
2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4])
+ (c^(1/4)*d*e^4*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^
2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d^2 + S
qrt[a]*e^2)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*
(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(
Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a
^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)*
Sqrt[a + c*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{4}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**4+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**4)**(3/2)*(d + e*x)), x)

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Mathematica [C]  time = 4.59494, size = 464, normalized size = 0.56 \[ \frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (\sqrt [4]{c} d \left (a e^5 \sqrt{a+c x^4} \log \left (e^2 x^2-d^2\right )-a e^5 \sqrt{a+c x^4} \log \left (\sqrt{a+c x^4} \sqrt{a e^4+c d^4}+a e^2+c d^2 x^2\right )+\sqrt{a e^4+c d^4} \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )\right )-2 \sqrt [4]{-1} a^{5/4} e^4 \sqrt{\frac{c x^4}{a}+1} \sqrt{a e^4+c d^4} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )\right )+c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} \left (\sqrt{a} e^2-i \sqrt{c} d^2\right ) \sqrt{a e^4+c d^4} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-\sqrt{a} c^{3/4} d^2 e^2 \sqrt{\frac{c x^4}{a}+1} \sqrt{a e^4+c d^4} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 a \sqrt [4]{c} d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left (a e^4+c d^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + c*x^4)^(3/2)),x]

[Out]

(-(Sqrt[a]*c^(3/4)*d^2*e^2*Sqrt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticE[I*A
rcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]) + c^(3/4)*d^2*((-I)*Sqrt[c]*d^2 + Sqrt
[a]*e^2)*Sqrt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(-2*(-1)^(1/4)*a^(5/4)*e^4*Sq
rt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2),
ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] + c^(1/4)*d*(Sqrt[c*d^4 + a*e^4]*(a*
e^3 + c*d*x*(d^2 - d*e*x + e^2*x^2)) + a*e^5*Sqrt[a + c*x^4]*Log[-d^2 + e^2*x^2]
- a*e^5*Sqrt[a + c*x^4]*Log[a*e^2 + c*d^2*x^2 + Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*
x^4]])))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^(1/4)*d*(c*d^4 + a*e^4)^(3/2)*Sqrt[a +
c*x^4])

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Maple [C]  time = 0.025, size = 496, normalized size = 0.6 \[ -2\,{c \left ( -1/4\,{\frac{d{e}^{2}{x}^{3}}{a \left ( a{e}^{4}+c{d}^{4} \right ) }}+1/4\,{\frac{{d}^{2}e{x}^{2}}{a \left ( a{e}^{4}+c{d}^{4} \right ) }}-1/4\,{\frac{{d}^{3}x}{a \left ( a{e}^{4}+c{d}^{4} \right ) }}-1/4\,{\frac{{e}^{3}}{ \left ( a{e}^{4}+c{d}^{4} \right ) c}} \right ){\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}+{\frac{c{d}^{3}}{2\,a \left ( a{e}^{4}+c{d}^{4} \right ) }\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}-{\frac{{\frac{i}{2}}{e}^{2}d}{a{e}^{4}+c{d}^{4}}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{{e}^{3}}{a{e}^{4}+c{d}^{4}} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{-i{e}^{2}}{{d}^{2}}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{-i\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^4+a)^(3/2),x)

[Out]

-2*c*(-1/4/a*e^2*d/(a*e^4+c*d^4)*x^3+1/4/a*d^2*e/(a*e^4+c*d^4)*x^2-1/4/a*d^3/(a*
e^4+c*d^4)*x-1/4*e^3/(a*e^4+c*d^4)/c)/((x^4+1/c*a)*c)^(1/2)+1/2*c/a*d^3/(a*e^4+c
*d^4)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(
1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-1/2*I*c
^(1/2)/a^(1/2)*e^2*d/(a*e^4+c*d^4)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2
)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*(EllipticF(x*(I/a^(
1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))+e^3/(a*e^4+c*d^
4)*(-1/2/(c*d^4/e^4+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a)/(c*d^4/e^4+a)^(1/
2)/(c*x^4+a)^(1/2))+1/(I/a^(1/2)*c^(1/2))^(1/2)/d*e*(1-I/a^(1/2)*c^(1/2)*x^2)^(1
/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1
/2))^(1/2),-I*a^(1/2)/c^(1/2)/d^2*e^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1
/2))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{4}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**4+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**4)**(3/2)*(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)),x, algorithm="giac")

[Out]

integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)), x)