3.1660 \(\int \frac{(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt{a+b x}}+\frac{32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt{a+b x}}-\frac{16 b \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{3 d^3}+\frac{40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac{4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}} \]

[Out]

(-4*(a + b*x)^(5/2))/(d*(c + d*x)^(1/4)) - (16*b*(b*c - a*d)*Sqrt[a + b*x]*(c +
d*x)^(3/4))/(3*d^3) + (40*b*(a + b*x)^(3/2)*(c + d*x)^(3/4))/(9*d^2) + (32*b^(1/
4)*(b*c - a*d)^(11/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/
4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*d^4*Sqrt[a + b*x]) - (32*b^(1/4)
*(b*c - a*d)^(11/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)
*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*d^4*Sqrt[a + b*x])

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Rubi [A]  time = 0.787367, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt{a+b x}}+\frac{32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt{a+b x}}-\frac{16 b \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{3 d^3}+\frac{40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac{4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/(c + d*x)^(5/4),x]

[Out]

(-4*(a + b*x)^(5/2))/(d*(c + d*x)^(1/4)) - (16*b*(b*c - a*d)*Sqrt[a + b*x]*(c +
d*x)^(3/4))/(3*d^3) + (40*b*(a + b*x)^(3/2)*(c + d*x)^(3/4))/(9*d^2) + (32*b^(1/
4)*(b*c - a*d)^(11/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/
4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*d^4*Sqrt[a + b*x]) - (32*b^(1/4)
*(b*c - a*d)^(11/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)
*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*d^4*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 101.099, size = 447, normalized size = 1.76 \[ - \frac{32 \sqrt [4]{b} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{11}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 d^{4} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{16 \sqrt [4]{b} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{11}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 d^{4} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{32 \sqrt{b} \sqrt [4]{c + d x} \left (a d - b c\right )^{\frac{3}{2}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{3 d^{3} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{40 b \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{4}}}{9 d^{2}} + \frac{16 b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )}{3 d^{3}} - \frac{4 \left (a + b x\right )^{\frac{5}{2}}}{d \sqrt [4]{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)/(d*x+c)**(5/4),x)

[Out]

-32*b**(1/4)*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/
sqrt(a*d - b*c) + 1)**2))*(a*d - b*c)**(11/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d -
b*c) + 1)*elliptic_e(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/
(3*d**4*sqrt(a - b*c/d + b*(c + d*x)/d)) + 16*b**(1/4)*sqrt((a*d - b*c + b*(c +
d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*(a*d - b*c)*
*(11/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_f(2*atan(b**(1/4)*(
c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(3*d**4*sqrt(a - b*c/d + b*(c + d*x)/d
)) + 32*sqrt(b)*(c + d*x)**(1/4)*(a*d - b*c)**(3/2)*sqrt(a - b*c/d + b*(c + d*x)
/d)/(3*d**3*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 40*b*(a + b*x)**(3/2)
*(c + d*x)**(3/4)/(9*d**2) + 16*b*sqrt(a + b*x)*(c + d*x)**(3/4)*(a*d - b*c)/(3*
d**3) - 4*(a + b*x)**(5/2)/(d*(c + d*x)**(1/4))

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Mathematica [C]  time = 0.294941, size = 131, normalized size = 0.52 \[ \frac{4 (c+d x)^{3/4} \left (d (a+b x) \left (-\frac{9 (b c-a d)^2}{c+d x}+b (4 a d-3 b c)+b^2 d x\right )+8 b (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{9 d^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/(c + d*x)^(5/4),x]

[Out]

(4*(c + d*x)^(3/4)*(d*(a + b*x)*(b*(-3*b*c + 4*a*d) + b^2*d*x - (9*(b*c - a*d)^2
)/(c + d*x)) + 8*b*(b*c - a*d)^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometr
ic2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(9*d^4*Sqrt[a + b*x])

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/(d*x+c)^(5/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(5/4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{2}}}{\left (c + d x\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)/(d*x+c)**(5/4),x)

[Out]

Integral((a + b*x)**(5/2)/(c + d*x)**(5/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(5/4),x, algorithm="giac")

[Out]

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