3.1676 \(\int \frac{1}{\sqrt{a+b x} (c+d x)^{9/4}} \, dx\)

Optimal. Leaf size=236 \[ \frac{12 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt{a+b x} (b c-a d)^{5/4}}-\frac{12 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt{a+b x} (b c-a d)^{5/4}}+\frac{12 b \sqrt{a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^2}+\frac{4 \sqrt{a+b x}}{5 (c+d x)^{5/4} (b c-a d)} \]

[Out]

(4*Sqrt[a + b*x])/(5*(b*c - a*d)*(c + d*x)^(5/4)) + (12*b*Sqrt[a + b*x])/(5*(b*c
- a*d)^2*(c + d*x)^(1/4)) - (12*b^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Elli
pticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d*(b*c - a*d)
^(5/4)*Sqrt[a + b*x]) + (12*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*(b*c - a*d)^(5/4
)*Sqrt[a + b*x])

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Rubi [A]  time = 0.696794, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{12 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt{a+b x} (b c-a d)^{5/4}}-\frac{12 b^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt{a+b x} (b c-a d)^{5/4}}+\frac{12 b \sqrt{a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^2}+\frac{4 \sqrt{a+b x}}{5 (c+d x)^{5/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x])/(5*(b*c - a*d)*(c + d*x)^(5/4)) + (12*b*Sqrt[a + b*x])/(5*(b*c
- a*d)^2*(c + d*x)^(1/4)) - (12*b^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Elli
pticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d*(b*c - a*d)
^(5/4)*Sqrt[a + b*x]) + (12*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*(b*c - a*d)^(5/4
)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 84.3008, size = 420, normalized size = 1.78 \[ \frac{12 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d \left (a d - b c\right )^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{6 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d \left (a d - b c\right )^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{12 b^{\frac{3}{2}} \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 \left (a d - b c\right )^{\frac{5}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{12 b \sqrt{a + b x}}{5 \sqrt [4]{c + d x} \left (a d - b c\right )^{2}} - \frac{4 \sqrt{a + b x}}{5 \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*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*(a*d - b*c)**(5/
4)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 6*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*(a*d - b*c)**(5/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 12*b*
*(3/2)*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(5*(a*d - b*c)**(5/2)*(s
qrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 12*b*sqrt(a + b*x)/(5*(c + d*x)**(1
/4)*(a*d - b*c)**2) - 4*sqrt(a + b*x)/(5*(c + d*x)**(5/4)*(a*d - b*c))

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Mathematica [C]  time = 0.217096, size = 115, normalized size = 0.49 \[ -\frac{4 \left (b^2 (c+d x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (a d-4 b c-3 b d x)\right )}{5 d \sqrt{a+b x} (c+d x)^{5/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(9/4)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(9/4)),x, algorithm="fricas")

[Out]

integral(1/((d^2*x^2 + 2*c*d*x + c^2)*sqrt(b*x + a)*(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(1/(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. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(9/4)),x, algorithm="giac")

[Out]

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