3.1725 \(\int \frac{(a+b x)^{7/4}}{(c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=730 \[ -\frac{7 \sqrt{b} \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{d^{5/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}-\frac{7 \sqrt [4]{b} (b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{7 \sqrt [4]{b} (b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{\sqrt{2} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{14 b (a+b x)^{3/4} (c+d x)^{3/4}}{3 d^2}-\frac{4 (a+b x)^{7/4}}{d \sqrt [4]{c+d x}} \]

[Out]

(-4*(a + b*x)^(7/4))/(d*(c + d*x)^(1/4)) + (14*b*(a + b*x)^(3/4)*(c + d*x)^(3/4)
)/(3*d^2) - (7*Sqrt[b]*Sqrt[(a + b*x)*(c + d*x)]*Sqrt[(b*c + a*d + 2*b*d*x)^2]*S
qrt[(a*d + b*(c + 2*d*x))^2])/(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^(1/4)*(b*c - a*d)^(5/2)*((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]*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])/(Sqrt[2]*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^(1/4)*(b*c
- a*d)^(5/2)*((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]*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])/(2*Sqrt[2]*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 = 1.65393, antiderivative size = 730, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{7 \sqrt{b} \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{d^{5/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}-\frac{7 \sqrt [4]{b} (b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{7 \sqrt [4]{b} (b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{\sqrt{2} d^{11/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{14 b (a+b x)^{3/4} (c+d x)^{3/4}}{3 d^2}-\frac{4 (a+b x)^{7/4}}{d \sqrt [4]{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(7/4)/(c + d*x)^(5/4),x]

[Out]

(-4*(a + b*x)^(7/4))/(d*(c + d*x)^(1/4)) + (14*b*(a + b*x)^(3/4)*(c + d*x)^(3/4)
)/(3*d^2) - (7*Sqrt[b]*Sqrt[(a + b*x)*(c + d*x)]*Sqrt[(b*c + a*d + 2*b*d*x)^2]*S
qrt[(a*d + b*(c + 2*d*x))^2])/(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^(1/4)*(b*c - a*d)^(5/2)*((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]*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])/(Sqrt[2]*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^(1/4)*(b*c
- a*d)^(5/2)*((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]*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])/(2*Sqrt[2]*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 = 154.84, size = 874, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/4)/(d*x+c)**(5/4),x)

[Out]

-7*sqrt(2)*b**(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*(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)**(5/2)*(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)/(2*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)*b**(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*(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)**(5/2)*(2*sqr
t(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)/(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(b)*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)/(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)) + 14*b*(a + b*x)**(3/4)*(c + d*x)**(3/4)/(3*d**2)
- 4*(a + b*x)**(7/4)/(d*(c + d*x)**(1/4))

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Mathematica [C]  time = 0.360694, size = 98, normalized size = 0.13 \[ \frac{2 (a+b x)^{3/4} (c+d x)^{3/4} \left (\frac{7 b \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{\left (\frac{d (a+b x)}{a d-b c}\right )^{3/4}}+\frac{-6 a d+7 b c+b d x}{c+d x}\right )}{3 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/4)/(c + d*x)^(5/4),x]

[Out]

(2*(a + b*x)^(3/4)*(c + d*x)^(3/4)*((7*b*c - 6*a*d + b*d*x)/(c + d*x) + (7*b*Hyp
ergeometric2F1[1/4, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*c)
+ a*d))^(3/4)))/(3*d^2)

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{4}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/4)/(d*x+c)^(5/4),x)

[Out]

int((b*x+a)^(7/4)/(d*x+c)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{4}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(5/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/4)/(d*x + c)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{7}{4}}}{{\left (d x + c\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(5/4),x, algorithm="fricas")

[Out]

integral((b*x + a)^(7/4)/(d*x + c)^(5/4), 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)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{4}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(5/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/4)/(d*x + c)^(5/4), x)