3.1690 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{7/4}} \, dx\)

Optimal. Leaf size=325 \[ \frac{5 d^{3/4} (b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.677848, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{5 d^{3/4} (b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Rubi in Sympy [A]  time = 57.0925, size = 377, normalized size = 1.16 \[ - \frac{4 \left (c + d x\right )^{\frac{5}{4}}}{3 b \left (a + b x\right )^{\frac{3}{4}}} + \frac{10 d \sqrt [4]{a + b x} \sqrt [4]{c + d x}}{3 b^{2}} - \frac{5 \sqrt{2} d^{\frac{3}{4}} \sqrt{\frac{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}}{\left (a d - b c\right )^{2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{3}{2}} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{3}{4}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{a c + b d x^{2} + x \left (a d + b c\right )}}{\sqrt{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{6 b^{\frac{9}{4}} \left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{3}{4}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(c + d*x)**(5/4)/(3*b*(a + b*x)**(3/4)) + 10*d*(a + b*x)**(1/4)*(c + d*x)**(1
/4)/(3*b**2) - 5*sqrt(2)*d**(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*(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)**(3/2)*(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)/(6*b**(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))

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Mathematica [C]  time = 0.184341, size = 95, normalized size = 0.29 \[ -\frac{2 \sqrt [4]{c+d x} \left (\frac{5 d (a+b x) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [4]{\frac{d (a+b x)}{a d-b c}}}-5 a d+2 b c-3 b d x\right )}{3 b^2 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out