3.1705 \(\int \frac{1}{(a+b x)^{9/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=760 \[ -\frac{2 \sqrt{2} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt{b c-a d} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{4 \sqrt{2} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt{b c-a d} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{8 d^{3/2} \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{5 \sqrt{b} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3 (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}+\frac{8 d (c+d x)^{3/4}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac{4 (c+d x)^{3/4}}{5 (a+b x)^{5/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(3/4))/(5*(b*c - a*d)*(a + b*x)^(5/4)) + (8*d*(c + d*x)^(3/4))/(5*
(b*c - a*d)^2*(a + b*x)^(1/4)) - (8*d^(3/2)*Sqrt[(a + b*x)*(c + d*x)]*Sqrt[(b*c
+ a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(5*Sqrt[b]*(b*c - a*d)^3*(a +
b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(
a + b*x)*(c + d*x)])/(b*c - a*d))) + (4*Sqrt[2]*d^(5/4)*((a + b*x)*(c + d*x))^(1
/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]*Sq
rt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]*EllipticE[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])/(5*b^(3/4)*Sq
rt[b*c - a*d]*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d +
b*(c + 2*d*x))^2]) - (2*Sqrt[2]*d^(5/4)*((a + b*x)*(c + d*x))^(1/4)*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])/(5*b^(3/4)*Sqrt[b*c - a*d]*(a
+ b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d + b*(c + 2*d*x))^2
])

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Rubi [A]  time = 1.63554, antiderivative size = 760, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{2 \sqrt{2} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt{b c-a d} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{4 \sqrt{2} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt{b c-a d} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{8 d^{3/2} \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{5 \sqrt{b} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3 (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}+\frac{8 d (c+d x)^{3/4}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac{4 (c+d x)^{3/4}}{5 (a+b x)^{5/4} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a + b*x)^(9/4)*(c + d*x)^(1/4)),x]

[Out]

(-4*(c + d*x)^(3/4))/(5*(b*c - a*d)*(a + b*x)^(5/4)) + (8*d*(c + d*x)^(3/4))/(5*
(b*c - a*d)^2*(a + b*x)^(1/4)) - (8*d^(3/2)*Sqrt[(a + b*x)*(c + d*x)]*Sqrt[(b*c
+ a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(5*Sqrt[b]*(b*c - a*d)^3*(a +
b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(
a + b*x)*(c + d*x)])/(b*c - a*d))) + (4*Sqrt[2]*d^(5/4)*((a + b*x)*(c + d*x))^(1
/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]*Sq
rt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]*EllipticE[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])/(5*b^(3/4)*Sq
rt[b*c - a*d]*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d +
b*(c + 2*d*x))^2]) - (2*Sqrt[2]*d^(5/4)*((a + b*x)*(c + d*x))^(1/4)*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])/(5*b^(3/4)*Sqrt[b*c - a*d]*(a
+ b*x)^(1/4)*(c + d*x)^(1/4)*(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 = 156.849, size = 896, normalized size = 1.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(9/4)/(d*x+c)**(1/4),x)

[Out]

8*d*(c + d*x)**(3/4)/(5*(a + b*x)**(1/4)*(a*d - b*c)**2) + 4*(c + d*x)**(3/4)/(5
*(a + b*x)**(5/4)*(a*d - b*c)) - 8*d**(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)*sqrt(a*c + b*d*x**2 + x*(a*d + b*c))*sqrt((a*d +
b*c + 2*b*d*x)**2)/(5*sqrt(b)*(a + b*x)**(1/4)*(c + d*x)**(1/4)*(a*d - b*c)**3*
(2*sqrt(b)*sqrt(d)*sqrt(a*c + b*d*x**2 + x*(a*d + b*c))/(a*d - b*c) + 1)*(a*d +
b*c + 2*b*d*x)) + 4*sqrt(2)*d**(5/4)*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))**(1/4)*sqrt((a
*d + b*c + 2*b*d*x)**2)*elliptic_e(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)/(5*b**(3/4)*(a + b*x)**(1/4)*
(c + d*x)**(1/4)*sqrt(a*d - b*c)*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)) - 2*sqrt(2)*d**(5/4)*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)*s
qrt(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))**(1/4)*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)/(
5*b**(3/4)*(a + b*x)**(1/4)*(c + d*x)**(1/4)*sqrt(a*d - b*c)*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 = 0.227483, size = 102, normalized size = 0.13 \[ -\frac{4 (c+d x)^{3/4} \left (4 d (a+b x) \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-9 a d+3 b (c-2 d x)\right )}{15 (a+b x)^{5/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(9/4)*(c + d*x)^(1/4)),x]

[Out]

(-4*(c + d*x)^(3/4)*(-9*a*d + 3*b*(c - 2*d*x) + 4*d*(a + b*x)*((d*(a + b*x))/(-(
b*c) + a*d))^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))
/(15*(b*c - a*d)^2*(a + b*x)^(5/4))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{9}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(9/4)/(d*x+c)^(1/4),x)

[Out]

int(1/(b*x+a)^(9/4)/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{9}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/4)*(d*x + c)^(1/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(9/4)*(d*x + c)^(1/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/4)*(d*x + c)^(1/4)),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)^(1/4)*(d*x + c)^(1/4)), 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(1/(b*x+a)**(9/4)/(d*x+c)**(1/4),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(1/((b*x + a)^(9/4)*(d*x + c)^(1/4)),x, algorithm="giac")

[Out]

Timed out