3.1658 \(\int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 \sqrt [4]{c+d x}}{\sqrt{a+b x} (b c-a d)}-\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a+b x} (b c-a d)^{3/4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.157032, antiderivative size = 111, 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 \[ -\frac{2 \sqrt [4]{c+d x}}{\sqrt{a+b x} (b c-a d)}-\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a+b x} (b c-a d)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.7952, size = 165, normalized size = 1.49 \[ \frac{2 \sqrt [4]{c + d x}}{\sqrt{a + b x} \left (a d - b c\right )} + \frac{\sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{\sqrt [4]{b} \left (a d - b c\right )^{\frac{3}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.109709, size = 81, normalized size = 0.73 \[ -\frac{2 \sqrt [4]{c+d x} \left (\sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+1\right )}{\sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

_______________________________________________________________________________________

Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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