3.1653 \(\int \frac{1}{(a+b x)^{3/2} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{2 (c+d x)^{3/4}}{\sqrt{a+b x} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(3/4))/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[-((d*(a + b*x))/(b*c
- a*d))]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^
(3/4)*(b*c - a*d)^(1/4)*Sqrt[a + b*x]) - (2*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*E
llipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*(b*c
- a*d)^(1/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.644875, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{2 (c+d x)^{3/4}}{\sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c + d*x)^(3/4))/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[-((d*(a + b*x))/(b*c
- a*d))]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^
(3/4)*(b*c - a*d)^(1/4)*Sqrt[a + b*x]) - (2*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*E
llipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*(b*c
- a*d)^(1/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 65.5061, size = 379, normalized size = 1.98 \[ \frac{2 \left (c + d x\right )^{\frac{3}{4}}}{\sqrt{a + b x} \left (a d - b c\right )} - \frac{2 d \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{\sqrt{b} \left (a d - b c\right )^{\frac{3}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{2 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt [4]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{\sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt [4]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c + d*x)**(3/4)/(sqrt(a + b*x)*(a*d - b*c)) - 2*d*(c + d*x)**(1/4)*sqrt(a - b
*c/d + b*(c + d*x)/d)/(sqrt(b)*(a*d - b*c)**(3/2)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*
d - b*c) + 1)) + 2*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))*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elli
ptic_e(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(b**(3/4)*(a*d
- b*c)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 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))*(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**(3/4)*(a*d - b*c)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.107385, size = 83, normalized size = 0.43 \[ \frac{2 (c+d x)^{3/4} \left (\sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-3\right )}{3 \sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c + d*x)^(3/4)*(-3 + Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/
2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(3*(b*c - a*d)*Sqrt[a + b*x])

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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