3.1644 \(\int \frac{(c+d x)^{5/4}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=144 \[ \frac{20 (b c-a d)^{9/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{21 b^{9/4} d \sqrt{a+b x}}+\frac{20 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b^2}+\frac{4 \sqrt{a+b x} (c+d x)^{5/4}}{7 b} \]

[Out]

(20*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/4))/(21*b^2) + (4*Sqrt[a + b*x]*(c +
d*x)^(5/4))/(7*b) + (20*(b*c - a*d)^(9/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ell
ipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(21*b^(9/4)*d*S
qrt[a + b*x])

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Rubi [A]  time = 0.207746, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{20 (b c-a d)^{9/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{21 b^{9/4} d \sqrt{a+b x}}+\frac{20 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b^2}+\frac{4 \sqrt{a+b x} (c+d x)^{5/4}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

(20*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/4))/(21*b^2) + (4*Sqrt[a + b*x]*(c +
d*x)^(5/4))/(7*b) + (20*(b*c - a*d)^(9/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ell
ipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(21*b^(9/4)*d*S
qrt[a + b*x])

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Rubi in Sympy [A]  time = 29.9499, size = 197, normalized size = 1.37 \[ \frac{4 \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{4}}}{7 b} - \frac{20 \sqrt{a + b x} \sqrt [4]{c + d x} \left (a d - b c\right )}{21 b^{2}} + \frac{10 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{9}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{9}{4}} d \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/4)/(b*x+a)**(1/2),x)

[Out]

4*sqrt(a + b*x)*(c + d*x)**(5/4)/(7*b) - 20*sqrt(a + b*x)*(c + d*x)**(1/4)*(a*d
- b*c)/(21*b**2) + 10*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)**(9/4)*(sqrt(b)*sqrt(c + d*x)/sqr
t(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)/(21*b**(9/4)*d*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.206123, size = 111, normalized size = 0.77 \[ \frac{4 \sqrt [4]{c+d x} \left (5 (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) (5 a d-8 b c-3 b d x)\right )}{21 b^2 d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c + d*x)^(1/4)*(-(d*(a + b*x)*(-8*b*c + 5*a*d - 3*b*d*x)) + 5*(b*c - a*d)^2*
Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*(c + d*x)
)/(b*c - a*d)]))/(21*b^2*d*Sqrt[a + b*x])

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/4)/(b*x+a)^(1/2),x)

[Out]

int((d*x+c)^(5/4)/(b*x+a)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(5/4)/(b*x+a)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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