3.1633 \(\int \frac{\sqrt [4]{c+d x}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{2 \sqrt [4]{b c-a d} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{5/4} \sqrt{a+b x}}-\frac{2 \sqrt [4]{c+d x}}{b \sqrt{a+b x}} \]

[Out]

(-2*(c + d*x)^(1/4))/(b*Sqrt[a + b*x]) + (2*(b*c - a*d)^(1/4)*Sqrt[-((d*(a + b*x
))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)],
-1])/(b^(5/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.1523, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt [4]{b c-a d} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{5/4} \sqrt{a+b x}}-\frac{2 \sqrt [4]{c+d x}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c + d*x)^(1/4))/(b*Sqrt[a + b*x]) + (2*(b*c - a*d)^(1/4)*Sqrt[-((d*(a + b*x
))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)],
-1])/(b^(5/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 20.5396, size = 160, normalized size = 1.54 \[ - \frac{2 \sqrt [4]{c + d x}}{b \sqrt{a + b x}} + \frac{\sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \sqrt [4]{a d - b c} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c + d*x)**(1/4)/(b*sqrt(a + b*x)) + sqrt((a*d - b*c + b*(c + d*x))/((a*d - b
*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*(a*d - b*c)**(1/4)*(sqrt(b)
*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_f(2*atan(b**(1/4)*(c + d*x)**(1/4)/
(a*d - b*c)**(1/4)), 1/2)/(b**(5/4)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.0920358, size = 74, normalized size = 0.71 \[ \frac{2 \sqrt [4]{c+d x} \left (\sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-1\right )}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c + d*x)^(1/4)*(-1 + Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/
4, 1/2, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(b*Sqrt[a + b*x])

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{dx+c} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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