3.1661 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=220 \[ \frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}} \]

[Out]

(-4*(a + b*x)^(3/2))/(d*(c + d*x)^(1/4)) + (24*b*Sqrt[a + b*x]*(c + d*x)^(3/4))/
(5*d^2) - (48*b^(1/4)*(b*c - a*d)^(7/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ellip
ticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a + b
*x]) + (48*b^(1/4)*(b*c - a*d)^(7/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Elliptic
F[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a + b*x]
)

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Rubi [A]  time = 0.685242, antiderivative size = 220, 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 \[ \frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(a + b*x)^(3/2))/(d*(c + d*x)^(1/4)) + (24*b*Sqrt[a + b*x]*(c + d*x)^(3/4))/
(5*d^2) - (48*b^(1/4)*(b*c - a*d)^(7/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ellip
ticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a + b
*x]) + (48*b^(1/4)*(b*c - a*d)^(7/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Elliptic
F[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^3*Sqrt[a + b*x]
)

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Rubi in Sympy [A]  time = 82.5275, size = 415, normalized size = 1.89 \[ - \frac{48 \sqrt [4]{b} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{24 \sqrt [4]{b} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{48 \sqrt{b} \sqrt [4]{c + d x} \sqrt{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 d^{2} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{24 b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{5 d^{2}} - \frac{4 \left (a + b x\right )^{\frac{3}{2}}}{d \sqrt [4]{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-48*b**(1/4)*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))*(a*d - b*c)**(7/4)*(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**3*sqrt(a - b*c/d + b*(c + d*x)/d)) + 24*b**(1/4)*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))*(a*d - b*c)**
(7/4)*(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)/(5*d**3*sqrt(a - b*c/d + b*(c + d*x)/d))
+ 48*sqrt(b)*(c + d*x)**(1/4)*sqrt(a*d - b*c)*sqrt(a - b*c/d + b*(c + d*x)/d)/(
5*d**2*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 24*b*sqrt(a + b*x)*(c + d*
x)**(3/4)/(5*d**2) - 4*(a + b*x)**(3/2)/(d*(c + d*x)**(1/4))

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Mathematica [C]  time = 0.311935, size = 98, normalized size = 0.45 \[ \frac{4 \sqrt{a+b x} (c+d x)^{3/4} \left (\frac{4 b \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+\frac{-5 a d+6 b c+b d x}{c+d x}\right )}{5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(3/4)*((6*b*c - 5*a*d + b*d*x)/(c + d*x) + (4*b*Hyper
geometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/Sqrt[(d*(a + b*x))/(-(b*
c) + a*d)]))/(5*d^2)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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