3.1673 \(\int \frac{(a+b x)^{5/2}}{(c+d x)^{9/4}} \, dx\)

Optimal. Leaf size=248 \[ \frac{96 b^{5/4} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^4 \sqrt{a+b x}}-\frac{96 b^{5/4} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^4 \sqrt{a+b x}}+\frac{48 b^2 \sqrt{a+b x} (c+d x)^{3/4}}{5 d^3}-\frac{8 b (a+b x)^{3/2}}{d^2 \sqrt [4]{c+d x}}-\frac{4 (a+b x)^{5/2}}{5 d (c+d x)^{5/4}} \]

[Out]

(-4*(a + b*x)^(5/2))/(5*d*(c + d*x)^(5/4)) - (8*b*(a + b*x)^(3/2))/(d^2*(c + d*x
)^(1/4)) + (48*b^2*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*d^3) - (96*b^(5/4)*(b*c - a
*d)^(7/4)*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])/(5*d^4*Sqrt[a + b*x]) + (96*b^(5/4)*(b*c - a*d)
^(7/4)*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])/(5*d^4*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.754715, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{96 b^{5/4} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^4 \sqrt{a+b x}}-\frac{96 b^{5/4} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^4 \sqrt{a+b x}}+\frac{48 b^2 \sqrt{a+b x} (c+d x)^{3/4}}{5 d^3}-\frac{8 b (a+b x)^{3/2}}{d^2 \sqrt [4]{c+d x}}-\frac{4 (a+b x)^{5/2}}{5 d (c+d x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/(c + d*x)^(9/4),x]

[Out]

(-4*(a + b*x)^(5/2))/(5*d*(c + d*x)^(5/4)) - (8*b*(a + b*x)^(3/2))/(d^2*(c + d*x
)^(1/4)) + (48*b^2*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*d^3) - (96*b^(5/4)*(b*c - a
*d)^(7/4)*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])/(5*d^4*Sqrt[a + b*x]) + (96*b^(5/4)*(b*c - a*d)
^(7/4)*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])/(5*d^4*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 100.779, size = 442, normalized size = 1.78 \[ - \frac{96 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{4} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{48 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{4} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{96 b^{\frac{3}{2}} \sqrt [4]{c + d x} \sqrt{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 d^{3} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{48 b^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{5 d^{3}} - \frac{8 b \left (a + b x\right )^{\frac{3}{2}}}{d^{2} \sqrt [4]{c + d x}} - \frac{4 \left (a + b x\right )^{\frac{5}{2}}}{5 d \left (c + d x\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)/(d*x+c)**(9/4),x)

[Out]

-96*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)**(7/4)*(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**4*sqrt(a - b*c/d + b*(c + d*x)/d)) + 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)**
(7/4)*(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**4*sqrt(a - b*c/d + b*(c + d*x)/d))
+ 96*b**(3/2)*(c + d*x)**(1/4)*sqrt(a*d - b*c)*sqrt(a - b*c/d + b*(c + d*x)/d)/
(5*d**3*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 48*b**2*sqrt(a + b*x)*(c
+ d*x)**(3/4)/(5*d**3) - 8*b*(a + b*x)**(3/2)/(d**2*(c + d*x)**(1/4)) - 4*(a + b
*x)**(5/2)/(5*d*(c + d*x)**(5/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.250912, size = 141, normalized size = 0.57 \[ \frac{4 (c+d x)^{3/4} \left (-8 b^2 (b c-a d) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-\frac{d (a+b x) \left (-12 b (c+d x) (b c-a d)+(b c-a d)^2-b^2 (c+d x)^2\right )}{(c+d x)^2}\right )}{5 d^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/(c + d*x)^(9/4),x]

[Out]

(4*(c + d*x)^(3/4)*(-((d*(a + b*x)*((b*c - a*d)^2 - 12*b*(b*c - a*d)*(c + d*x) -
b^2*(c + d*x)^2))/(c + d*x)^2) - 8*b^2*(b*c - a*d)*Sqrt[(d*(a + b*x))/(-(b*c) +
a*d)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(5*d^4*Sqrt
[a + b*x])

_______________________________________________________________________________________

Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/(d*x+c)^(9/4),x)

[Out]

int((b*x+a)^(5/2)/(d*x+c)^(9/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(9/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/2)/(d*x + c)^(9/4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(9/4),x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)/((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)**(5/2)/(d*x+c)**(9/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(9/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/2)/(d*x + c)^(9/4), x)