3.1678 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{9/4}} \, dx\)

Optimal. Leaf size=303 \[ \frac{77 b^{5/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 \sqrt{a+b x} (b c-a d)^{13/4}}-\frac{77 b^{5/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 \sqrt{a+b x} (b c-a d)^{13/4}}+\frac{77 b d^2 \sqrt{a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^4}+\frac{77 d^2 \sqrt{a+b x}}{15 (c+d x)^{5/4} (b c-a d)^3}+\frac{11 d}{3 \sqrt{a+b x} (c+d x)^{5/4} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{5/4} (b c-a d)} \]

[Out]

-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/4)) + (11*d)/(3*(b*c - a*d)^2*Sqr
t[a + b*x]*(c + d*x)^(5/4)) + (77*d^2*Sqrt[a + b*x])/(15*(b*c - a*d)^3*(c + d*x)
^(5/4)) + (77*b*d^2*Sqrt[a + b*x])/(5*(b*c - a*d)^4*(c + d*x)^(1/4)) - (77*b^(5/
4)*d*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])/(5*(b*c - a*d)^(13/4)*Sqrt[a + b*x]) + (77*b^(5/4)*d
*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*(b*c - a*d)^(13/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.861511, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{77 b^{5/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 \sqrt{a+b x} (b c-a d)^{13/4}}-\frac{77 b^{5/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 \sqrt{a+b x} (b c-a d)^{13/4}}+\frac{77 b d^2 \sqrt{a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^4}+\frac{77 d^2 \sqrt{a+b x}}{15 (c+d x)^{5/4} (b c-a d)^3}+\frac{11 d}{3 \sqrt{a+b x} (c+d x)^{5/4} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{5/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(9/4)),x]

[Out]

-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/4)) + (11*d)/(3*(b*c - a*d)^2*Sqr
t[a + b*x]*(c + d*x)^(5/4)) + (77*d^2*Sqrt[a + b*x])/(15*(b*c - a*d)^3*(c + d*x)
^(5/4)) + (77*b*d^2*Sqrt[a + b*x])/(5*(b*c - a*d)^4*(c + d*x)^(1/4)) - (77*b^(5/
4)*d*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])/(5*(b*c - a*d)^(13/4)*Sqrt[a + b*x]) + (77*b^(5/4)*d
*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*(b*c - a*d)^(13/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 126.415, size = 490, normalized size = 1.62 \[ \frac{77 b^{\frac{5}{4}} d \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 \left (a d - b c\right )^{\frac{13}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{77 b^{\frac{5}{4}} d \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{10 \left (a d - b c\right )^{\frac{13}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{77 b^{\frac{3}{2}} d^{2} \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 \left (a d - b c\right )^{\frac{9}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{77 b d^{2} \sqrt{a + b x}}{5 \sqrt [4]{c + d x} \left (a d - b c\right )^{4}} - \frac{77 d^{2} \sqrt{a + b x}}{15 \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )^{3}} + \frac{11 d}{3 \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )^{2}} + \frac{2}{3 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(9/4),x)

[Out]

77*b**(5/4)*d*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_
e(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(5*(a*d - b*c)**(13
/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 77*b**(5/4)*d*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)/(10*(a*d - b*c)**(13/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 7
7*b**(3/2)*d**2*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(5*(a*d - b*c)*
*(9/2)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 77*b*d**2*sqrt(a + b*x)/(5
*(c + d*x)**(1/4)*(a*d - b*c)**4) - 77*d**2*sqrt(a + b*x)/(15*(c + d*x)**(5/4)*(
a*d - b*c)**3) + 11*d/(3*sqrt(a + b*x)*(c + d*x)**(5/4)*(a*d - b*c)**2) + 2/(3*(
a + b*x)**(3/2)*(c + d*x)**(5/4)*(a*d - b*c))

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Mathematica [C]  time = 0.382015, size = 156, normalized size = 0.51 \[ \frac{(c+d x)^{3/4} \left (-77 b^2 d \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-\frac{10 b^2 (b c-a d)}{a+b x}+\frac{156 b d^2 (a+b x)}{c+d x}+\frac{12 d^2 (a+b x) (b c-a d)}{(c+d x)^2}+75 b^2 d\right )}{15 \sqrt{a+b x} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(9/4)),x]

[Out]

((c + d*x)^(3/4)*(75*b^2*d - (10*b^2*(b*c - a*d))/(a + b*x) + (12*d^2*(b*c - a*d
)*(a + b*x))/(c + d*x)^2 + (156*b*d^2*(a + b*x))/(c + d*x) - 77*b^2*d*Sqrt[(d*(a
+ b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a
*d)]))/(15*(b*c - a*d)^4*Sqrt[a + b*x])

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/(d*x+c)^(9/4),x)

[Out]

int(1/(b*x+a)^(5/2)/(d*x+c)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(9/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(9/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} d^{2} x^{4} + a^{2} c^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(9/4)),x, algorithm="fricas")

[Out]

integral(1/((b^2*d^2*x^4 + a^2*c^2 + 2*(b^2*c*d + a*b*d^2)*x^3 + (b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*x^2 + 2*(a*b*c^2 + a^2*c*d)*x)*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)**(5/2)/(d*x+c)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.550249, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(9/4)),x, algorithm="giac")

[Out]

Done