3.2528 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=709 \[ -\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(((-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*A
rcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sq
rt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(c^(3/4)*(c*d^2 - b*d*e + a*e^2)^
(3/4)*(a + b*x + c*x^2)^(3/4))) - ((-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c)))^(3/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2
*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/
(c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(3/4)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)
*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 -
4*a*c)))^(3/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e
+ a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*c*(c*
d^2 - b*d*e + a*e^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)*(2*c*
d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
)))^(3/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^
2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*c*(c*d^2 - b
*d*e + a*e^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))

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Rubi [A]  time = 4.64139, antiderivative size = 709, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591 \[ -\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\sqrt{e} \left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{c^{3/4} \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c (b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(((-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*A
rcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sq
rt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(c^(3/4)*(c*d^2 - b*d*e + a*e^2)^
(3/4)*(a + b*x + c*x^2)^(3/4))) - ((-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c)))^(3/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2
*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/
(c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(3/4)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)
*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 -
4*a*c)))^(3/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e
+ a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*c*(c*
d^2 - b*d*e + a*e^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)*(2*c*
d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
)))^(3/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^
2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*c*(c*d^2 - b
*d*e + a*e^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.306478, size = 180, normalized size = 0.25 \[ -\frac{\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},\frac{3}{4};\frac{5}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{3 \sqrt{2} e (a+x (b+c x))^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4)*((e*(b + Sqrt[b^2 -
4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4)*AppellF1[3/2, 3/4, 3/4, 5/2, (2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c
*d + 2*c*e*x)])/(3*Sqrt[2]*e*(a + x*(b + c*x))^(3/4))

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Maple [F]  time = 0.133, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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