3.1664 \(\int \frac{1}{(a+b x)^{3/2} (c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{6 d \sqrt{a+b x}}{\sqrt [4]{c+d x} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)}-\frac{6 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{5/4}}+\frac{6 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{5/4}} \]

[Out]

-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/4)) - (6*d*Sqrt[a + b*x])/((b*c - a*d
)^2*(c + d*x)^(1/4)) + (6*b^(1/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[A
rcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/((b*c - a*d)^(5/4)*Sqrt
[a + b*x]) - (6*b^(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*c - a*d)^(5/4)*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.678642, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{6 d \sqrt{a+b x}}{\sqrt [4]{c+d x} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)}-\frac{6 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{5/4}}+\frac{6 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(3/2)*(c + d*x)^(5/4)),x]

[Out]

-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/4)) - (6*d*Sqrt[a + b*x])/((b*c - a*d
)^2*(c + d*x)^(1/4)) + (6*b^(1/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[A
rcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/((b*c - a*d)^(5/4)*Sqrt
[a + b*x]) - (6*b^(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*c - a*d)^(5/4)*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 83.1945, size = 410, normalized size = 1.85 \[ - \frac{6 \sqrt [4]{b} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{\left (a d - b c\right )^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{3 \sqrt [4]{b} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{\left (a d - b c\right )^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{6 \sqrt{b} d \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{\left (a d - b c\right )^{\frac{5}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} - \frac{6 d \sqrt{a + b x}}{\sqrt [4]{c + d x} \left (a d - b c\right )^{2}} + \frac{2}{\sqrt{a + b x} \sqrt [4]{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(5/4),x)

[Out]

-6*b**(1/4)*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/s
qrt(a*d - b*c) + 1)**2))*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_e(
2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/((a*d - b*c)**(5/4)*s
qrt(a - b*c/d + b*(c + d*x)/d)) + 3*b**(1/4)*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)/((a*d - b*c)**(5/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) + 6*sqrt(b)*d*(
c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/((a*d - b*c)**(5/2)*(sqrt(b)*sqr
t(c + d*x)/sqrt(a*d - b*c) + 1)) - 6*d*sqrt(a + b*x)/((c + d*x)**(1/4)*(a*d - b*
c)**2) + 2/(sqrt(a + b*x)*(c + d*x)**(1/4)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [C]  time = 0.242213, size = 99, normalized size = 0.45 \[ \frac{2 b (c+d x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-4 a d-2 b (c+3 d x)}{\sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(5/4)),x]

[Out]

(-4*a*d - 2*b*(c + 3*d*x) + 2*b*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*(c + d*x)*Hyp
ergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*Sqrt[a
+ b*x]*(c + d*x)^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(3/2)/(d*x+c)^(5/4),x)

[Out]

int(1/(b*x+a)^(3/2)/(d*x+c)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/4)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/4)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(3/2)/(d*x+c)**(5/4),x)

[Out]

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(5/4)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/4)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/4)), x)