3.1659 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=149 \[ \frac{5 d \sqrt [4]{c+d x}}{3 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}+\frac{5 d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} \sqrt{a+b x} (b c-a d)^{7/4}} \]

[Out]

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

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Rubi [A]  time = 0.190031, antiderivative size = 149, 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 \[ \frac{5 d \sqrt [4]{c+d x}}{3 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt [4]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)}+\frac{5 d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} \sqrt{a+b x} (b c-a d)^{7/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 29.9737, size = 202, normalized size = 1.36 \[ \frac{5 d \sqrt [4]{c + d x}}{3 \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \sqrt [4]{c + d x}}{3 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{5 d \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{6 \sqrt [4]{b} \left (a d - b c\right )^{\frac{7}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(3/4),x)

[Out]

5*d*(c + d*x)**(1/4)/(3*sqrt(a + b*x)*(a*d - b*c)**2) + 2*(c + d*x)**(1/4)/(3*(a
+ b*x)**(3/2)*(a*d - b*c)) + 5*d*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(s
qrt(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)
/(6*b**(1/4)*(a*d - b*c)**(7/4)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.207412, size = 102, normalized size = 0.68 \[ \frac{\sqrt [4]{c+d x} \left (5 d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+7 a d-2 b c+5 b d x\right )}{3 (a+b x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(3/4)),x]

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/(d*x+c)^(3/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/4)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/4)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/2)/(d*x+c)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/4)),x, algorithm="giac")

[Out]

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