3.1675 \(\int \frac{\sqrt{a+b x}}{(c+d x)^{9/4}} \, dx\)

Optimal. Leaf size=232 \[ \frac{8 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{8 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{8 b \sqrt{a+b x}}{5 d \sqrt [4]{c+d x} (b c-a d)}-\frac{4 \sqrt{a+b x}}{5 d (c+d x)^{5/4}} \]

[Out]

(-4*Sqrt[a + b*x])/(5*d*(c + d*x)^(5/4)) + (8*b*Sqrt[a + b*x])/(5*d*(b*c - a*d)*
(c + d*x)^(1/4)) - (8*b^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSi
n[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^2*(b*c - a*d)^(1/4)*Sq
rt[a + b*x]) + (8*b^(5/4)*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])/(5*d^2*(b*c - a*d)^(1/4)*Sqrt[a
+ b*x])

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Rubi [A]  time = 0.694057, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{8 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{8 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{8 b \sqrt{a+b x}}{5 d \sqrt [4]{c+d x} (b c-a d)}-\frac{4 \sqrt{a+b x}}{5 d (c+d x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/(c + d*x)^(9/4),x]

[Out]

(-4*Sqrt[a + b*x])/(5*d*(c + d*x)^(5/4)) + (8*b*Sqrt[a + b*x])/(5*d*(b*c - a*d)*
(c + d*x)^(1/4)) - (8*b^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSi
n[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^2*(b*c - a*d)^(1/4)*Sq
rt[a + b*x]) + (8*b^(5/4)*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])/(5*d^2*(b*c - a*d)^(1/4)*Sqrt[a
+ b*x])

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Rubi in Sympy [A]  time = 82.6812, size = 420, normalized size = 1.81 \[ - \frac{8 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{2} \sqrt [4]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{4 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{2} \sqrt [4]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{8 b^{\frac{3}{2}} \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 d \left (a d - b c\right )^{\frac{3}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} - \frac{8 b \sqrt{a + b x}}{5 d \sqrt [4]{c + d x} \left (a d - b c\right )} - \frac{4 \sqrt{a + b x}}{5 d \left (c + d x\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/(d*x+c)**(9/4),x)

[Out]

-8*b**(5/4)*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/s
qrt(a*d - b*c) + 1)**2))*(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)/(5*d**2*(a*d - b*c)**
(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) + 4*b**(5/4)*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))*(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)/(5*d**2*(a*d - b*c)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) +
8*b**(3/2)*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(5*d*(a*d - b*c)**(
3/2)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) - 8*b*sqrt(a + b*x)/(5*d*(c +
d*x)**(1/4)*(a*d - b*c)) - 4*sqrt(a + b*x)/(5*d*(c + d*x)**(5/4))

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Mathematica [C]  time = 0.218899, size = 116, normalized size = 0.5 \[ \frac{8 b^2 (c+d x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-12 d (a+b x) (a d+b (c+2 d x))}{15 d^2 \sqrt{a+b x} (c+d x)^{5/4} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(c + d*x)^(9/4),x]

[Out]

(-12*d*(a + b*x)*(a*d + b*(c + 2*d*x)) + 8*b^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)
]*(c + d*x)^2*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/(15*d
^2*(-(b*c) + a*d)*Sqrt[a + b*x]*(c + d*x)^(5/4))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/(d*x+c)^(9/4),x)

[Out]

int((b*x+a)^(1/2)/(d*x+c)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(9/4),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x + a)/(d*x + c)^(9/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x + a}}{{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\left (d x + c\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(9/4),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)/((d^2*x^2 + 2*c*d*x + c^2)*(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)**(1/2)/(d*x+c)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(9/4),x, algorithm="giac")

[Out]

integrate(sqrt(b*x + a)/(d*x + c)^(9/4), x)