3.1650 \(\int \frac{(a+b x)^{3/2}}{\sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{16 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt{a+b x}}+\frac{16 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{15 d^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d} \]

[Out]

(-8*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/4))/(15*d^2) + (4*(a + b*x)^(3/2)*(c
+ d*x)^(3/4))/(9*d) + (16*(b*c - a*d)^(11/4)*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])/(15*b^(3/4)*
d^3*Sqrt[a + b*x]) - (16*(b*c - a*d)^(11/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*E
llipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(15*b^(3/4)*d
^3*Sqrt[a + b*x])

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Rubi [A]  time = 0.722679, antiderivative size = 229, 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 \[ -\frac{16 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt{a+b x}}+\frac{16 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{15 d^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d} \]

Antiderivative was successfully verified.

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

[Out]

(-8*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/4))/(15*d^2) + (4*(a + b*x)^(3/2)*(c
+ d*x)^(3/4))/(9*d) + (16*(b*c - a*d)^(11/4)*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])/(15*b^(3/4)*
d^3*Sqrt[a + b*x]) - (16*(b*c - a*d)^(11/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*E
llipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(15*b^(3/4)*d
^3*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 83.5821, size = 422, normalized size = 1.84 \[ \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{4}}}{9 d} + \frac{8 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )}{15 d^{2}} + \frac{16 \sqrt [4]{c + d x} \left (a d - b c\right )^{\frac{3}{2}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{15 \sqrt{b} d^{2} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} - \frac{16 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{11}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{8 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{11}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(a + b*x)**(3/2)*(c + d*x)**(3/4)/(9*d) + 8*sqrt(a + b*x)*(c + d*x)**(3/4)*(a*
d - b*c)/(15*d**2) + 16*(c + d*x)**(1/4)*(a*d - b*c)**(3/2)*sqrt(a - b*c/d + b*(
c + d*x)/d)/(15*sqrt(b)*d**2*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) - 16*s
qrt((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)**(11/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elli
ptic_e(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(15*b**(3/4)*d
**3*sqrt(a - b*c/d + b*(c + d*x)/d)) + 8*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)**(11/4)*(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)/(15*b**(3/4)*d**3*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.20255, size = 107, normalized size = 0.47 \[ \frac{4 (c+d x)^{3/4} \left (4 (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (11 a d-6 b c+5 b d x)\right )}{45 d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c + d*x)^(3/4)*(d*(a + b*x)*(-6*b*c + 11*a*d + 5*b*d*x) + 4*(b*c - a*d)^2*Sq
rt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/
(b*c - a*d)]))/(45*d^3*Sqrt[a + b*x])

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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