3.1674 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{9/4}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{48 b^{5/4} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}+\frac{48 b^{5/4} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{24 b \sqrt{a+b x}}{5 d^2 \sqrt [4]{c+d x}}-\frac{4 (a+b x)^{3/2}}{5 d (c+d x)^{5/4}} \]

[Out]

(-4*(a + b*x)^(3/2))/(5*d*(c + d*x)^(5/4)) - (24*b*Sqrt[a + b*x])/(5*d^2*(c + d*
x)^(1/4)) + (48*b^(5/4)*(b*c - a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ell
ipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a +
b*x]) - (48*b^(5/4)*(b*c - a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ellipt
icF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a + b*
x])

_______________________________________________________________________________________

Rubi [A]  time = 0.677457, 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 \[ -\frac{48 b^{5/4} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}+\frac{48 b^{5/4} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{24 b \sqrt{a+b x}}{5 d^2 \sqrt [4]{c+d x}}-\frac{4 (a+b x)^{3/2}}{5 d (c+d x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(c + d*x)^(9/4),x]

[Out]

(-4*(a + b*x)^(3/2))/(5*d*(c + d*x)^(5/4)) - (24*b*Sqrt[a + b*x])/(5*d^2*(c + d*
x)^(1/4)) + (48*b^(5/4)*(b*c - a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ell
ipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a +
b*x]) - (48*b^(5/4)*(b*c - a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ellipt
icF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a + b*
x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 80.9538, size = 416, normalized size = 1.87 \[ - \frac{48 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{3}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{24 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{3}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{48 b^{\frac{3}{2}} \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 d^{2} \sqrt{a d - b c} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} - \frac{24 b \sqrt{a + b x}}{5 d^{2} \sqrt [4]{c + d x}} - \frac{4 \left (a + b x\right )^{\frac{3}{2}}}{5 d \left (c + d x\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/(d*x+c)**(9/4),x)

[Out]

-48*b**(5/4)*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))*(a*d - b*c)**(3/4)*(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)/(
5*d**3*sqrt(a - b*c/d + b*(c + d*x)/d)) + 24*b**(5/4)*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))*(a*d - b*c)**
(3/4)*(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)/(5*d**3*sqrt(a - b*c/d + b*(c + d*x)/d))
+ 48*b**(3/2)*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(5*d**2*sqrt(a*d
- b*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) - 24*b*sqrt(a + b*x)/(5*d**
2*(c + d*x)**(1/4)) - 4*(a + b*x)**(3/2)/(5*d*(c + d*x)**(5/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.219551, size = 107, normalized size = 0.48 \[ \frac{16 b^2 (c+d x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-4 d (a+b x) (a d+6 b c+7 b d x)}{5 d^3 \sqrt{a+b x} (c+d x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(c + d*x)^(9/4),x]

[Out]

(-4*d*(a + b*x)*(6*b*c + a*d + 7*b*d*x) + 16*b^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*
d)]*(c + d*x)^2*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/(5*
d^3*Sqrt[a + b*x]*(c + d*x)^(5/4))

_______________________________________________________________________________________

Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/(d*x+c)^(9/4),x)

[Out]

int((b*x+a)^(3/2)/(d*x+c)^(9/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(9/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(9/4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(9/4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/(d*x+c)**(9/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(9/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(9/4), x)