3.3028 \(\int \frac{1}{(a+b x) \sqrt{c+d x} (e+f x)^{3/4}} \, dx\)

Optimal. Leaf size=252 \[ -\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt [4]{d} \sqrt{c+d x} (b e-a f)}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt [4]{d} \sqrt{c+d x} (b e-a f)} \]

[Out]

(-2*(d*e - c*f)^(1/4)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[-((Sqrt[b]*S
qrt[d*e - c*f])/(Sqrt[d]*Sqrt[b*e - a*f])), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*
e - c*f)^(1/4)], -1])/(d^(1/4)*(b*e - a*f)*Sqrt[c + d*x]) - (2*(d*e - c*f)^(1/4)
*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[(Sqrt[b]*Sqrt[d*e - c*f])/(Sqrt[d
]*Sqrt[b*e - a*f]), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e - c*f)^(1/4)], -1])/(d
^(1/4)*(b*e - a*f)*Sqrt[c + d*x])

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Rubi [A]  time = 0.958862, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt [4]{d} \sqrt{c+d x} (b e-a f)}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt [4]{d} \sqrt{c+d x} (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d*e - c*f)^(1/4)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[-((Sqrt[b]*S
qrt[d*e - c*f])/(Sqrt[d]*Sqrt[b*e - a*f])), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*
e - c*f)^(1/4)], -1])/(d^(1/4)*(b*e - a*f)*Sqrt[c + d*x]) - (2*(d*e - c*f)^(1/4)
*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[(Sqrt[b]*Sqrt[d*e - c*f])/(Sqrt[d
]*Sqrt[b*e - a*f]), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e - c*f)^(1/4)], -1])/(d
^(1/4)*(b*e - a*f)*Sqrt[c + d*x])

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

[Out]

Timed out

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Mathematica [C]  time = 0.918661, size = 271, normalized size = 1.08 \[ -\frac{36 d f (a+b x) F_1\left (\frac{5}{4};\frac{1}{2},\frac{3}{4};\frac{9}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{5 b \sqrt{c+d x} (e+f x)^{3/4} \left (9 d f (a+b x) F_1\left (\frac{5}{4};\frac{1}{2},\frac{3}{4};\frac{9}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(3 a d f-3 b d e) F_1\left (\frac{9}{4};\frac{1}{2},\frac{7}{4};\frac{13}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+2 f (a d-b c) F_1\left (\frac{9}{4};\frac{3}{2},\frac{3}{4};\frac{13}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-36*d*f*(a + b*x)*AppellF1[5/4, 1/2, 3/4, 9/4, (-(b*c) + a*d)/(d*(a + b*x)), (-
(b*e) + a*f)/(f*(a + b*x))])/(5*b*Sqrt[c + d*x]*(e + f*x)^(3/4)*(9*d*f*(a + b*x)
*AppellF1[5/4, 1/2, 3/4, 9/4, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a
+ b*x))] + (-3*b*d*e + 3*a*d*f)*AppellF1[9/4, 1/2, 7/4, 13/4, (-(b*c) + a*d)/(d
*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))] + 2*(-(b*c) + a*d)*f*AppellF1[9/4, 3/
2, 3/4, 13/4, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))]))

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{1}{bx+a}{\frac{1}{\sqrt{dx+c}}} \left ( fx+e \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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