3.1648 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{9/2}} \, dx\)

Optimal. Leaf size=213 \[ \frac{5 d^3 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{84 b^{9/4} \sqrt{a+b x} (b c-a d)^{7/4}}+\frac{5 d^3 \sqrt [4]{c+d x}}{84 b^2 \sqrt{a+b x} (b c-a d)^2}-\frac{d^2 \sqrt [4]{c+d x}}{42 b^2 (a+b x)^{3/2} (b c-a d)}-\frac{d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac{2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}} \]

[Out]

-(d*(c + d*x)^(1/4))/(7*b^2*(a + b*x)^(5/2)) - (d^2*(c + d*x)^(1/4))/(42*b^2*(b*
c - a*d)*(a + b*x)^(3/2)) + (5*d^3*(c + d*x)^(1/4))/(84*b^2*(b*c - a*d)^2*Sqrt[a
+ b*x]) - (2*(c + d*x)^(5/4))/(7*b*(a + b*x)^(7/2)) + (5*d^3*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])/(84*b^(9/4)*(b*c - a*d)^(7/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.305429, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{5 d^3 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{84 b^{9/4} \sqrt{a+b x} (b c-a d)^{7/4}}+\frac{5 d^3 \sqrt [4]{c+d x}}{84 b^2 \sqrt{a+b x} (b c-a d)^2}-\frac{d^2 \sqrt [4]{c+d x}}{42 b^2 (a+b x)^{3/2} (b c-a d)}-\frac{d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac{2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(c + d*x)^(1/4))/(7*b^2*(a + b*x)^(5/2)) - (d^2*(c + d*x)^(1/4))/(42*b^2*(b*
c - a*d)*(a + b*x)^(3/2)) + (5*d^3*(c + d*x)^(1/4))/(84*b^2*(b*c - a*d)^2*Sqrt[a
+ b*x]) - (2*(c + d*x)^(5/4))/(7*b*(a + b*x)^(7/2)) + (5*d^3*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])/(84*b^(9/4)*(b*c - a*d)^(7/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 52.0886, size = 260, normalized size = 1.22 \[ - \frac{2 \left (c + d x\right )^{\frac{5}{4}}}{7 b \left (a + b x\right )^{\frac{7}{2}}} + \frac{5 d^{3} \sqrt [4]{c + d x}}{84 b^{2} \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{d^{2} \sqrt [4]{c + d x}}{42 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{d \sqrt [4]{c + d x}}{7 b^{2} \left (a + b x\right )^{\frac{5}{2}}} + \frac{5 d^{3} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{168 b^{\frac{9}{4}} \left (a d - b c\right )^{\frac{7}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c + d*x)**(5/4)/(7*b*(a + b*x)**(7/2)) + 5*d**3*(c + d*x)**(1/4)/(84*b**2*sq
rt(a + b*x)*(a*d - b*c)**2) + d**2*(c + d*x)**(1/4)/(42*b**2*(a + b*x)**(3/2)*(a
*d - b*c)) - d*(c + d*x)**(1/4)/(7*b**2*(a + b*x)**(5/2)) + 5*d**3*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)/(168*b**(9/4)*(a*d - b*c)**(7/4)*sqrt(a - b*c/
d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.334209, size = 181, normalized size = 0.85 \[ \frac{\sqrt [4]{c+d x} \left (-5 a^3 d^3-a^2 b d^2 (2 c+17 d x)+a b^2 d \left (36 c^2+68 c d x+17 d^2 x^2\right )+5 d^3 (a+b x)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+b^3 \left (-\left (24 c^3+36 c^2 d x+2 c d^2 x^2-5 d^3 x^3\right )\right )\right )}{84 b^2 (a+b x)^{7/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(1/4)*(-5*a^3*d^3 - a^2*b*d^2*(2*c + 17*d*x) + a*b^2*d*(36*c^2 + 68*c
*d*x + 17*d^2*x^2) - b^3*(24*c^3 + 36*c^2*d*x + 2*c*d^2*x^2 - 5*d^3*x^3) + 5*d^3
*(a + b*x)^3*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/4, 1/2, 5/4,
(b*(c + d*x))/(b*c - a*d)]))/(84*b^2*(b*c - a*d)^2*(a + b*x)^(7/2))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{9}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt{b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)^(5/4)/((b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a
^4)*sqrt(b*x + a)), x)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.680455, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done