3.1687 \(\int (a+b x)^{5/4} (c+d x)^{5/4} \, dx\)

Optimal. Leaf size=408 \[ \frac{5 (b c-a d)^{9/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} b^{9/4} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{5 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{84 b^2 d^2}+\frac{(a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)^2}{42 b^2 d}+\frac{(a+b x)^{9/4} \sqrt [4]{c+d x} (b c-a d)}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b} \]

[Out]

(-5*(b*c - a*d)^3*(a + b*x)^(1/4)*(c + d*x)^(1/4))/(84*b^2*d^2) + ((b*c - a*d)^2
*(a + b*x)^(5/4)*(c + d*x)^(1/4))/(42*b^2*d) + ((b*c - a*d)*(a + b*x)^(9/4)*(c +
d*x)^(1/4))/(7*b^2) + (2*(a + b*x)^(9/4)*(c + d*x)^(5/4))/(7*b) + (5*(b*c - a*d
)^(9/2)*((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])/(168*Sqrt[2]*b^(9/4)*d^(9/4)*(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])

_______________________________________________________________________________________

Rubi [A]  time = 1.17692, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 (b c-a d)^{9/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} b^{9/4} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{5 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{84 b^2 d^2}+\frac{(a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)^2}{42 b^2 d}+\frac{(a+b x)^{9/4} \sqrt [4]{c+d x} (b c-a d)}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*(b*c - a*d)^3*(a + b*x)^(1/4)*(c + d*x)^(1/4))/(84*b^2*d^2) + ((b*c - a*d)^2
*(a + b*x)^(5/4)*(c + d*x)^(1/4))/(42*b^2*d) + ((b*c - a*d)*(a + b*x)^(9/4)*(c +
d*x)^(1/4))/(7*b^2) + (2*(a + b*x)^(9/4)*(c + d*x)^(5/4))/(7*b) + (5*(b*c - a*d
)^(9/2)*((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])/(168*Sqrt[2]*b^(9/4)*d^(9/4)*(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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 89.1864, size = 449, normalized size = 1.1 \[ \frac{2 \left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{9}{4}}}{7 d} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{9}{4}} \left (a d - b c\right )}{7 d^{2}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )^{2}}{42 b d^{2}} - \frac{5 \sqrt [4]{a + b x} \sqrt [4]{c + d x} \left (a d - b c\right )^{3}}{84 b^{2} d^{2}} + \frac{5 \sqrt{2} \sqrt{\frac{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}}{\left (a d - b c\right )^{2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{9}{2}} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{3}{4}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{a c + b d x^{2} + x \left (a d + b c\right )}}{\sqrt{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{336 b^{\frac{9}{4}} d^{\frac{9}{4}} \left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{3}{4}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a + b*x)**(5/4)*(c + d*x)**(9/4)/(7*d) + (a + b*x)**(1/4)*(c + d*x)**(9/4)*(a
*d - b*c)/(7*d**2) + (a + b*x)**(1/4)*(c + d*x)**(5/4)*(a*d - b*c)**2/(42*b*d**2
) - 5*(a + b*x)**(1/4)*(c + d*x)**(1/4)*(a*d - b*c)**3/(84*b**2*d**2) + 5*sqrt(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*(2*sqrt(b)*sqrt(d)*sqrt(a*c + b*d*x**2 + x*(a*d + b*c))/(a*d - b*c) + 1)*
*2))*(a*d - b*c)**(9/2)*(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)/(336*b**(9/4)*d**(9/4)*(a + b*x)**(3/4)*(c +
d*x)**(3/4)*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*b*d*x))

_______________________________________________________________________________________

Mathematica [C]  time = 0.245064, size = 183, normalized size = 0.45 \[ \frac{\sqrt [4]{c+d x} \left (5 (b c-a d)^4 \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) \left (5 a^3 d^3-a^2 b d^2 (17 c+2 d x)-a b^2 d \left (17 c^2+68 c d x+36 d^2 x^2\right )+b^3 \left (5 c^3-2 c^2 d x-36 c d^2 x^2-24 d^3 x^3\right )\right )\right )}{84 b^2 d^3 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(1/4)*(-(d*(a + b*x)*(5*a^3*d^3 - a^2*b*d^2*(17*c + 2*d*x) - a*b^2*d*
(17*c^2 + 68*c*d*x + 36*d^2*x^2) + b^3*(5*c^3 - 2*c^2*d*x - 36*c*d^2*x^2 - 24*d^
3*x^3))) + 5*(b*c - a*d)^4*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*Hypergeometric2F
1[1/4, 3/4, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(84*b^2*d^3*(a + b*x)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{5}{4}}} \left ( dx+c \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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((b*x + a)^(5/4)*(d*x + c)^(5/4),x, algorithm="giac")

[Out]

Timed out