3.1471 \(\int (a+b x)^{5/2} (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=227 \[ -\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{7/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{64 b^2 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{80 b^2 d}+\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b} \]

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^2*d^3) - ((b*c - a*d)^3*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(64*b^2*d^2) + ((b*c - a*d)^2*(a + b*x)^(5/2)*Sqrt[c
+ d*x])/(80*b^2*d) + (3*(b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(40*b^2) + (
(a + b*x)^(7/2)*(c + d*x)^(3/2))/(5*b) - (3*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[
a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(7/2))

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Rubi [A]  time = 0.305819, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{7/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{64 b^2 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{80 b^2 d}+\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^2*d^3) - ((b*c - a*d)^3*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(64*b^2*d^2) + ((b*c - a*d)^2*(a + b*x)^(5/2)*Sqrt[c
+ d*x])/(80*b^2*d) + (3*(b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(40*b^2) + (
(a + b*x)^(7/2)*(c + d*x)^(3/2))/(5*b) - (3*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[
a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(7/2))

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Rubi in Sympy [A]  time = 46.7164, size = 197, normalized size = 0.87 \[ \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{5 d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{8 d^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}}{16 d^{3}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}}{64 b d^{3}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4}}{128 b^{2} d^{3}} + \frac{3 \left (a d - b c\right )^{5} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{5}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(5/2)*(c + d*x)**(5/2)/(5*d) + (a + b*x)**(3/2)*(c + d*x)**(5/2)*(a*d
- b*c)/(8*d**2) + sqrt(a + b*x)*(c + d*x)**(5/2)*(a*d - b*c)**2/(16*d**3) + sqr
t(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**3/(64*b*d**3) - 3*sqrt(a + b*x)*sqrt(c
+ d*x)*(a*d - b*c)**4/(128*b**2*d**3) + 3*(a*d - b*c)**5*atanh(sqrt(d)*sqrt(a +
b*x)/(sqrt(b)*sqrt(c + d*x)))/(128*b**(5/2)*d**(7/2))

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Mathematica [A]  time = 0.242925, size = 233, normalized size = 1.03 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-15 a^4 d^4+10 a^3 b d^3 (7 c+d x)+2 a^2 b^2 d^2 \left (64 c^2+233 c d x+124 d^2 x^2\right )+2 a b^3 d \left (-35 c^3+23 c^2 d x+256 c d^2 x^2+168 d^3 x^3\right )+b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )}{640 b^2 d^3}-\frac{3 (b c-a d)^5 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{5/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-15*a^4*d^4 + 10*a^3*b*d^3*(7*c + d*x) + 2*a^2*b^2
*d^2*(64*c^2 + 233*c*d*x + 124*d^2*x^2) + 2*a*b^3*d*(-35*c^3 + 23*c^2*d*x + 256*
c*d^2*x^2 + 168*d^3*x^3) + b^4*(15*c^4 - 10*c^3*d*x + 8*c^2*d^2*x^2 + 176*c*d^3*
x^3 + 128*d^4*x^4)))/(640*b^2*d^3) - (3*(b*c - a*d)^5*Log[b*c + a*d + 2*b*d*x +
2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(256*b^(5/2)*d^(7/2))

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Maple [B]  time = 0.011, size = 853, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5/d*(b*x+a)^(5/2)*(d*x+c)^(5/2)+1/8/d*(b*x+a)^(3/2)*(d*x+c)^(5/2)*a+1/16/d*(b*
x+a)^(1/2)*(d*x+c)^(5/2)*a^2-1/8/d^2*(b*x+a)^(1/2)*(d*x+c)^(5/2)*a*b*c+3/64/d^2*
(d*x+c)^(3/2)*(b*x+a)^(1/2)*a*c^2*b+3/32/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3*c+3/3
2/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^3*b-15/256*d/b*((b*x+a)*(d*x+c))^(1/2)/(d*
x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*
c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^4*c+15/128*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/
(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(
1/2))/(b*d)^(1/2)*a^3*c^2+15/256/d^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+
a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))
/(b*d)^(1/2)*a*c^4*b^2-3/64/d*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^2*c-1/64/d^3*(d*x+c)
^(3/2)*(b*x+a)^(1/2)*c^3*b^2-3/128*d/b^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^4-9/64/d*
(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2*c^2-3/128/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^4*b^
2-1/8/d^2*(b*x+a)^(3/2)*(d*x+c)^(5/2)*b*c+1/16/d^3*(b*x+a)^(1/2)*(d*x+c)^(5/2)*b
^2*c^2+1/64/b*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^3+3/256*d^2/b^2*((b*x+a)*(d*x+c))^(1
/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+
(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^5-15/128/d*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)
^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x
+a*c)^(1/2))/(b*d)^(1/2)*a^2*c^3*b-3/256/d^3*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/
2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c
)^(1/2))/(b*d)^(1/2)*c^5*b^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25509, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (128 \, b^{4} d^{4} x^{4} + 15 \, b^{4} c^{4} - 70 \, a b^{3} c^{3} d + 128 \, a^{2} b^{2} c^{2} d^{2} + 70 \, a^{3} b c d^{3} - 15 \, a^{4} d^{4} + 16 \,{\left (11 \, b^{4} c d^{3} + 21 \, a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (b^{4} c^{2} d^{2} + 64 \, a b^{3} c d^{3} + 31 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{3} d - 23 \, a b^{3} c^{2} d^{2} - 233 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{2560 \, \sqrt{b d} b^{2} d^{3}}, \frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} + 15 \, b^{4} c^{4} - 70 \, a b^{3} c^{3} d + 128 \, a^{2} b^{2} c^{2} d^{2} + 70 \, a^{3} b c d^{3} - 15 \, a^{4} d^{4} + 16 \,{\left (11 \, b^{4} c d^{3} + 21 \, a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (b^{4} c^{2} d^{2} + 64 \, a b^{3} c d^{3} + 31 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{3} d - 23 \, a b^{3} c^{2} d^{2} - 233 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{1280 \, \sqrt{-b d} b^{2} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(4*(128*b^4*d^4*x^4 + 15*b^4*c^4 - 70*a*b^3*c^3*d + 128*a^2*b^2*c^2*d^2
+ 70*a^3*b*c*d^3 - 15*a^4*d^4 + 16*(11*b^4*c*d^3 + 21*a*b^3*d^4)*x^3 + 8*(b^4*c^
2*d^2 + 64*a*b^3*c*d^3 + 31*a^2*b^2*d^4)*x^2 - 2*(5*b^4*c^3*d - 23*a*b^3*c^2*d^2
- 233*a^2*b^2*c*d^3 - 5*a^3*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 1
5*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c
*d^4 - a^5*d^5)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x +
c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*
sqrt(b*d)))/(sqrt(b*d)*b^2*d^3), 1/1280*(2*(128*b^4*d^4*x^4 + 15*b^4*c^4 - 70*a*
b^3*c^3*d + 128*a^2*b^2*c^2*d^2 + 70*a^3*b*c*d^3 - 15*a^4*d^4 + 16*(11*b^4*c*d^3
+ 21*a*b^3*d^4)*x^3 + 8*(b^4*c^2*d^2 + 64*a*b^3*c*d^3 + 31*a^2*b^2*d^4)*x^2 - 2
*(5*b^4*c^3*d - 23*a*b^3*c^2*d^2 - 233*a^2*b^2*c*d^3 - 5*a^3*b*d^4)*x)*sqrt(-b*d
)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2
- 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*arctan(1/2*(2*b*d*x + b*c + a*d
)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^2*d^3)]

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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)**(5/2)*(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.353726, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done