3.1693 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{19/4}} \, dx\)

Optimal. Leaf size=401 \[ \frac{4 \sqrt{2} d^{15/4} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{231 b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (b c-a d)^{3/2} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{8 d^3 \sqrt [4]{c+d x}}{231 b^2 (a+b x)^{3/4} (b c-a d)^2}-\frac{4 d^2 \sqrt [4]{c+d x}}{231 b^2 (a+b x)^{7/4} (b c-a d)}-\frac{4 d \sqrt [4]{c+d x}}{33 b^2 (a+b x)^{11/4}}-\frac{4 (c+d x)^{5/4}}{15 b (a+b x)^{15/4}} \]

[Out]

(-4*d*(c + d*x)^(1/4))/(33*b^2*(a + b*x)^(11/4)) - (4*d^2*(c + d*x)^(1/4))/(231*
b^2*(b*c - a*d)*(a + b*x)^(7/4)) + (8*d^3*(c + d*x)^(1/4))/(231*b^2*(b*c - a*d)^
2*(a + b*x)^(3/4)) - (4*(c + d*x)^(5/4))/(15*b*(a + b*x)^(15/4)) + (4*Sqrt[2]*d^
(15/4)*((a + b*x)*(c + d*x))^(3/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[b]
*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/((
b*c - a*d)^2*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]
*EllipticF[2*ArcTan[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt[b
*c - a*d]], 1/2])/(231*b^(9/4)*(b*c - a*d)^(3/2)*(a + b*x)^(3/4)*(c + d*x)^(3/4)
*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d + b*(c + 2*d*x))^2])

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Rubi [A]  time = 1.01361, antiderivative size = 401, 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 \[ \frac{4 \sqrt{2} d^{15/4} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{231 b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (b c-a d)^{3/2} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{8 d^3 \sqrt [4]{c+d x}}{231 b^2 (a+b x)^{3/4} (b c-a d)^2}-\frac{4 d^2 \sqrt [4]{c+d x}}{231 b^2 (a+b x)^{7/4} (b c-a d)}-\frac{4 d \sqrt [4]{c+d x}}{33 b^2 (a+b x)^{11/4}}-\frac{4 (c+d x)^{5/4}}{15 b (a+b x)^{15/4}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(c + d*x)^(5/4)/(a + b*x)^(19/4),x]

[Out]

(-4*d*(c + d*x)^(1/4))/(33*b^2*(a + b*x)^(11/4)) - (4*d^2*(c + d*x)^(1/4))/(231*
b^2*(b*c - a*d)*(a + b*x)^(7/4)) + (8*d^3*(c + d*x)^(1/4))/(231*b^2*(b*c - a*d)^
2*(a + b*x)^(3/4)) - (4*(c + d*x)^(5/4))/(15*b*(a + b*x)^(15/4)) + (4*Sqrt[2]*d^
(15/4)*((a + b*x)*(c + d*x))^(3/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[b]
*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/((
b*c - a*d)^2*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]
*EllipticF[2*ArcTan[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt[b
*c - a*d]], 1/2])/(231*b^(9/4)*(b*c - a*d)^(3/2)*(a + b*x)^(3/4)*(c + d*x)^(3/4)
*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d + b*(c + 2*d*x))^2])

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Rubi in Sympy [A]  time = 88.0072, size = 447, normalized size = 1.11 \[ - \frac{4 \left (c + d x\right )^{\frac{5}{4}}}{15 b \left (a + b x\right )^{\frac{15}{4}}} + \frac{8 d^{3} \sqrt [4]{c + d x}}{231 b^{2} \left (a + b x\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} + \frac{4 d^{2} \sqrt [4]{c + d x}}{231 b^{2} \left (a + b x\right )^{\frac{7}{4}} \left (a d - b c\right )} - \frac{4 d \sqrt [4]{c + d x}}{33 b^{2} \left (a + b x\right )^{\frac{11}{4}}} + \frac{4 \sqrt{2} d^{\frac{15}{4}} \sqrt{\frac{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}}{\left (a d - b c\right )^{2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right )^{2}}} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{3}{4}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{a c + b d x^{2} + x \left (a d + b c\right )}}{\sqrt{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} \left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )^{\frac{3}{2}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/4)/(b*x+a)**(19/4),x)

[Out]

-4*(c + d*x)**(5/4)/(15*b*(a + b*x)**(15/4)) + 8*d**3*(c + d*x)**(1/4)/(231*b**2
*(a + b*x)**(3/4)*(a*d - b*c)**2) + 4*d**2*(c + d*x)**(1/4)/(231*b**2*(a + b*x)*
*(7/4)*(a*d - b*c)) - 4*d*(c + d*x)**(1/4)/(33*b**2*(a + b*x)**(11/4)) + 4*sqrt(
2)*d**(15/4)*sqrt((b*d*(4*a*c + 4*b*d*x**2 + x*(4*a*d + 4*b*c)) + (a*d - b*c)**2
)/((a*d - b*c)**2*(2*sqrt(b)*sqrt(d)*sqrt(a*c + b*d*x**2 + x*(a*d + b*c))/(a*d -
b*c) + 1)**2))*(2*sqrt(b)*sqrt(d)*sqrt(a*c + b*d*x**2 + x*(a*d + b*c))/(a*d - b
*c) + 1)*(a*c + b*d*x**2 + x*(a*d + b*c))**(3/4)*sqrt((a*d + b*c + 2*b*d*x)**2)*
elliptic_f(2*atan(sqrt(2)*b**(1/4)*d**(1/4)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1
/4)/sqrt(a*d - b*c)), 1/2)/(231*b**(9/4)*(a + b*x)**(3/4)*(c + d*x)**(3/4)*(a*d
- b*c)**(3/2)*sqrt(b*d*(4*a*c + 4*b*d*x**2 + x*(4*a*d + 4*b*c)) + (a*d - b*c)**2
)*(a*d + b*c + 2*b*d*x))

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Mathematica [C]  time = 0.328222, size = 179, normalized size = 0.45 \[ \frac{4 \sqrt [4]{c+d x} \left (-20 a^3 d^3-12 a^2 b d^2 (c+6 d x)+a b^2 d \left (119 c^2+214 c d x+35 d^2 x^2\right )+20 d^3 (a+b x)^3 \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+b^3 \left (-\left (77 c^3+112 c^2 d x+5 c d^2 x^2-10 d^3 x^3\right )\right )\right )}{1155 b^2 (a+b x)^{15/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/4)/(a + b*x)^(19/4),x]

[Out]

(4*(c + d*x)^(1/4)*(-20*a^3*d^3 - 12*a^2*b*d^2*(c + 6*d*x) + a*b^2*d*(119*c^2 +
214*c*d*x + 35*d^2*x^2) - b^3*(77*c^3 + 112*c^2*d*x + 5*c*d^2*x^2 - 10*d^3*x^3)
+ 20*d^3*(a + b*x)^3*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*Hypergeometric2F1[1/4,
3/4, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(1155*b^2*(b*c - a*d)^2*(a + b*x)^(15/4)
)

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{19}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/4)/(b*x+a)^(19/4),x)

[Out]

int((d*x+c)^(5/4)/(b*x+a)^(19/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{19}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(19/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(19/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(19/4),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)*(b*x + a)^(3/4)), 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)**(19/4),x)

[Out]

Timed out

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GIAC/XCAS [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)^(19/4),x, algorithm="giac")

[Out]

Timed out