3.1667 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{7/4}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{16 b^{3/4} (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^3 \sqrt{a+b x}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}} \]

[Out]

(-4*(a + b*x)^(3/2))/(3*d*(c + d*x)^(3/4)) + (8*b*Sqrt[a + b*x]*(c + d*x)^(1/4))
/(3*d^2) - (16*b^(3/4)*(b*c - a*d)^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Elli
pticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*d^3*Sqrt[a +
b*x])

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Rubi [A]  time = 0.20091, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{16 b^{3/4} (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^3 \sqrt{a+b x}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(c + d*x)^(7/4),x]

[Out]

(-4*(a + b*x)^(3/2))/(3*d*(c + d*x)^(3/4)) + (8*b*Sqrt[a + b*x]*(c + d*x)^(1/4))
/(3*d^2) - (16*b^(3/4)*(b*c - a*d)^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Elli
pticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*d^3*Sqrt[a +
b*x])

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Rubi in Sympy [A]  time = 29.6633, size = 194, normalized size = 1.42 \[ \frac{8 b^{\frac{3}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{5}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{8 b \sqrt{a + b x} \sqrt [4]{c + d x}}{3 d^{2}} - \frac{4 \left (a + b x\right )^{\frac{3}{2}}}{3 d \left (c + d x\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/(d*x+c)**(7/4),x)

[Out]

8*b**(3/4)*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sq
rt(a*d - b*c) + 1)**2))*(a*d - b*c)**(5/4)*(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)/(3*
d**3*sqrt(a - b*c/d + b*(c + d*x)/d)) + 8*b*sqrt(a + b*x)*(c + d*x)**(1/4)/(3*d*
*2) - 4*(a + b*x)**(3/2)/(3*d*(c + d*x)**(3/4))

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Mathematica [C]  time = 0.354895, size = 98, normalized size = 0.72 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x} \left (\frac{4 b \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+\frac{-a d+2 b c+b d x}{c+d x}\right )}{3 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(c + d*x)^(7/4),x]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4)*((2*b*c - a*d + b*d*x)/(c + d*x) + (4*b*Hyperge
ometric2F1[1/4, 1/2, 5/4, (b*(c + d*x))/(b*c - a*d)])/Sqrt[(d*(a + b*x))/(-(b*c)
+ a*d)]))/(3*d^2)

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/(d*x+c)^(7/4),x)

[Out]

int((b*x+a)^(3/2)/(d*x+c)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(7/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(7/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{7}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(7/4),x, algorithm="fricas")

[Out]

integral((b*x + a)^(3/2)/(d*x + c)^(7/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}}}{\left (c + d x\right )^{\frac{7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/(d*x+c)**(7/4),x)

[Out]

Integral((a + b*x)**(3/2)/(c + d*x)**(7/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(7/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(7/4), x)