3.1642 \(\int (a+b x)^{3/2} (c+d x)^{5/4} \, dx\)

Optimal. Leaf size=220 \[ \frac{16 (b c-a d)^{17/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{231 b^2 d^2}+\frac{4 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac{4 (a+b x)^{5/2} \sqrt [4]{c+d x} (b c-a d)}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b} \]

[Out]

(-8*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(1/4))/(231*b^2*d^2) + (4*(b*c - a*d)^
2*(a + b*x)^(3/2)*(c + d*x)^(1/4))/(231*b^2*d) + (4*(b*c - a*d)*(a + b*x)^(5/2)*
(c + d*x)^(1/4))/(33*b^2) + (4*(a + b*x)^(5/2)*(c + d*x)^(5/4))/(15*b) + (16*(b*
c - a*d)^(17/4)*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])/(231*b^(9/4)*d^3*Sqrt[a + b*x])

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Rubi [A]  time = 0.416716, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{16 (b c-a d)^{17/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{231 b^2 d^2}+\frac{4 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac{4 (a+b x)^{5/2} \sqrt [4]{c+d x} (b c-a d)}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b} \]

Antiderivative was successfully verified.

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

[Out]

(-8*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(1/4))/(231*b^2*d^2) + (4*(b*c - a*d)^
2*(a + b*x)^(3/2)*(c + d*x)^(1/4))/(231*b^2*d) + (4*(b*c - a*d)*(a + b*x)^(5/2)*
(c + d*x)^(1/4))/(33*b^2) + (4*(a + b*x)^(5/2)*(c + d*x)^(5/4))/(15*b) + (16*(b*
c - a*d)^(17/4)*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])/(231*b^(9/4)*d^3*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 54.4552, size = 269, normalized size = 1.22 \[ \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{9}{4}}}{15 d} + \frac{8 \sqrt{a + b x} \left (c + d x\right )^{\frac{9}{4}} \left (a d - b c\right )}{55 d^{2}} + \frac{16 \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )^{2}}{385 b d^{2}} - \frac{16 \sqrt{a + b x} \sqrt [4]{c + d x} \left (a d - b c\right )^{3}}{231 b^{2} d^{2}} + \frac{8 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{17}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(a + b*x)**(3/2)*(c + d*x)**(9/4)/(15*d) + 8*sqrt(a + b*x)*(c + d*x)**(9/4)*(a
*d - b*c)/(55*d**2) + 16*sqrt(a + b*x)*(c + d*x)**(5/4)*(a*d - b*c)**2/(385*b*d*
*2) - 16*sqrt(a + b*x)*(c + d*x)**(1/4)*(a*d - b*c)**3/(231*b**2*d**2) + 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)**(17/4)*(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)/(231*b**(9/4)*d**3
*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.366982, size = 182, normalized size = 0.83 \[ \frac{4 \sqrt [4]{c+d x} \left (20 (b c-a d)^4 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) \left (20 a^3 d^3-12 a^2 b d^2 (6 c+d x)-a b^2 d \left (35 c^2+214 c d x+119 d^2 x^2\right )+b^3 \left (10 c^3-5 c^2 d x-112 c d^2 x^2-77 d^3 x^3\right )\right )\right )}{1155 b^2 d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c + d*x)^(1/4)*(-(d*(a + b*x)*(20*a^3*d^3 - 12*a^2*b*d^2*(6*c + d*x) - a*b^2
*d*(35*c^2 + 214*c*d*x + 119*d^2*x^2) + b^3*(10*c^3 - 5*c^2*d*x - 112*c*d^2*x^2
- 77*d^3*x^3))) + 20*(b*c - a*d)^4*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeome
tric2F1[1/4, 1/2, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(1155*b^2*d^3*Sqrt[a + b*x])

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{{\frac{5}{4}}}\, dx \]

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. \[ \int{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{4}}\,{d x} \]

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. \[{\rm integral}\left ({\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

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)