3.1701 \(\int \frac{(a+b x)^{7/4}}{\sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=751 \[ \frac{7 (b c-a d)^{7/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} b^{3/4} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{7 (b c-a d)^{7/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{10 \sqrt{2} b^{3/4} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{7 (b c-a d) \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{10 \sqrt{b} d^{5/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}-\frac{7 (a+b x)^{3/4} (c+d x)^{3/4} (b c-a d)}{15 d^2}+\frac{2 (a+b x)^{7/4} (c+d x)^{3/4}}{5 d} \]

[Out]

(-7*(b*c - a*d)*(a + b*x)^(3/4)*(c + d*x)^(3/4))/(15*d^2) + (2*(a + b*x)^(7/4)*(
c + d*x)^(3/4))/(5*d) + (7*(b*c - a*d)*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])/(10*Sqrt[b]*d^(5/2)*(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))) - (7*(b*c - a*d)^(7/2)*((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]*Sq
rt[(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])/(10*Sqrt[2]*b^(3/4)*d
^(11/4)*(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]) + (7*(b*c - a*d)^(7/2)*((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))*Sq
rt[(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])/(20*Sqrt[2]*b^(3/4)*d^(11/4)*(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 = 1.72266, antiderivative size = 751, 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 \[ \frac{7 (b c-a d)^{7/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} b^{3/4} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{7 (b c-a d)^{7/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{10 \sqrt{2} b^{3/4} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{7 (b c-a d) \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{10 \sqrt{b} d^{5/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}-\frac{7 (a+b x)^{3/4} (c+d x)^{3/4} (b c-a d)}{15 d^2}+\frac{2 (a+b x)^{7/4} (c+d x)^{3/4}}{5 d} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(7/4)/(c + d*x)^(1/4),x]

[Out]

(-7*(b*c - a*d)*(a + b*x)^(3/4)*(c + d*x)^(3/4))/(15*d^2) + (2*(a + b*x)^(7/4)*(
c + d*x)^(3/4))/(5*d) + (7*(b*c - a*d)*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])/(10*Sqrt[b]*d^(5/2)*(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))) - (7*(b*c - a*d)^(7/2)*((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]*Sq
rt[(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])/(10*Sqrt[2]*b^(3/4)*d
^(11/4)*(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]) + (7*(b*c - a*d)^(7/2)*((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))*Sq
rt[(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])/(20*Sqrt[2]*b^(3/4)*d^(11/4)*(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 = 156.368, size = 889, normalized size = 1.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/4)/(d*x+c)**(1/4),x)

[Out]

2*(a + b*x)**(7/4)*(c + d*x)**(3/4)/(5*d) + 7*(a + b*x)**(3/4)*(c + d*x)**(3/4)*
(a*d - b*c)/(15*d**2) + 7*(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)*sqrt(a*c + b*d*x**2 + x*(a*d + b*c))*sqrt((a*d + b*c +
2*b*d*x)**2)/(10*sqrt(b)*d**(5/2)*(a + b*x)**(1/4)*(c + d*x)**(1/4)*(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)) - 7*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)**(7/2)*(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)/(20*b**(3/4)*d**(11/4)*(a + b*
x)**(1/4)*(c + d*x)**(1/4)*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)) + 7*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)**(7/2)*(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_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)/(40*b**(3/4)*d**(11/4)*(a + b*x)**(1/4)*(c + d*x)**(1/4)*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))

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Mathematica [C]  time = 0.221917, size = 107, normalized size = 0.14 \[ \frac{(c+d x)^{3/4} \left (7 (b c-a d)^2 \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (13 a d-7 b c+6 b d x)\right )}{15 d^3 \sqrt [4]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/4)/(c + d*x)^(1/4),x]

[Out]

((c + d*x)^(3/4)*(d*(a + b*x)*(-7*b*c + 13*a*d + 6*b*d*x) + 7*(b*c - a*d)^2*((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*d^3*(a + b*x)^(1/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/4)/(d*x+c)^(1/4),x)

[Out]

int((b*x+a)^(7/4)/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(1/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/4)/(d*x + c)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{7}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(1/4),x, algorithm="fricas")

[Out]

integral((b*x + a)^(7/4)/(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((b*x+a)**(7/4)/(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((b*x + a)^(7/4)/(d*x + c)^(1/4),x, algorithm="giac")

[Out]

Timed out