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

Optimal. Leaf size=376 \[ -\frac{\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{11/2} d^{7/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \]

[Out]

((b*c - a*d)^3*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(512*b^5*d^3) + ((b*c - a*d)^2*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*(a + b*x)
^(3/2)*Sqrt[c + d*x])/(256*b^5*d^2) + ((b*c - a*d)*(5*b^2*c^2 + 14*a*b*c*d + 21*
a^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(192*b^4*d^2) + ((5*b^2*c^2 + 14*a*b*c
*d + 21*a^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(160*b^3*d^2) - ((5*b*c + 9*a*
d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^
(7/2))/(6*b*d) - ((b*c - a*d)^4*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(S
qrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(11/2)*d^(7/2))

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Rubi [A]  time = 0.818609, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{11/2} d^{7/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^3*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(512*b^5*d^3) + ((b*c - a*d)^2*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*(a + b*x)
^(3/2)*Sqrt[c + d*x])/(256*b^5*d^2) + ((b*c - a*d)*(5*b^2*c^2 + 14*a*b*c*d + 21*
a^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(192*b^4*d^2) + ((5*b^2*c^2 + 14*a*b*c
*d + 21*a^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(160*b^3*d^2) - ((5*b*c + 9*a*
d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^
(7/2))/(6*b*d) - ((b*c - a*d)^4*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(S
qrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(11/2)*d^(7/2))

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Rubi in Sympy [A]  time = 75.1212, size = 360, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{2}}}{6 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{2}} \left (9 a d + 5 b c\right )}{60 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (21 a^{2} d^{2} + 14 a b c d + 5 b^{2} c^{2}\right )}{160 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (21 a^{2} d^{2} + 14 a b c d + 5 b^{2} c^{2}\right )}{192 b^{4} d^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (21 a^{2} d^{2} + 14 a b c d + 5 b^{2} c^{2}\right )}{256 b^{5} d^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (21 a^{2} d^{2} + 14 a b c d + 5 b^{2} c^{2}\right )}{512 b^{5} d^{3}} - \frac{\left (a d - b c\right )^{4} \left (21 a^{2} d^{2} + 14 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{512 b^{\frac{11}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(d*x+c)**(5/2)*(b*x+a)**(1/2),x)

[Out]

x*(a + b*x)**(3/2)*(c + d*x)**(7/2)/(6*b*d) - (a + b*x)**(3/2)*(c + d*x)**(7/2)*
(9*a*d + 5*b*c)/(60*b**2*d**2) + (a + b*x)**(3/2)*(c + d*x)**(5/2)*(21*a**2*d**2
+ 14*a*b*c*d + 5*b**2*c**2)/(160*b**3*d**2) - (a + b*x)**(3/2)*(c + d*x)**(3/2)
*(a*d - b*c)*(21*a**2*d**2 + 14*a*b*c*d + 5*b**2*c**2)/(192*b**4*d**2) + (a + b*
x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**2*(21*a**2*d**2 + 14*a*b*c*d + 5*b**2*c**2)
/(256*b**5*d**2) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3*(21*a**2*d**2 + 14
*a*b*c*d + 5*b**2*c**2)/(512*b**5*d**3) - (a*d - b*c)**4*(21*a**2*d**2 + 14*a*b*
c*d + 5*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(512*b**
(11/2)*d**(7/2))

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Mathematica [A]  time = 0.318285, size = 320, normalized size = 0.85 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (315 a^5 d^5-105 a^4 b d^4 (9 c+2 d x)+2 a^3 b^2 d^3 \left (419 c^2+308 c d x+84 d^2 x^2\right )-2 a^2 b^3 d^2 \left (45 c^3+262 c^2 d x+244 c d^2 x^2+72 d^3 x^3\right )+a b^4 d \left (-65 c^4+40 c^3 d x+408 c^2 d^2 x^2+416 c d^3 x^3+128 d^4 x^4\right )+5 b^5 \left (15 c^5-10 c^4 d x+8 c^3 d^2 x^2+432 c^2 d^3 x^3+640 c d^4 x^4+256 d^5 x^5\right )\right )}{7680 b^5 d^3}-\frac{(b c-a d)^4 \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{1024 b^{11/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(315*a^5*d^5 - 105*a^4*b*d^4*(9*c + 2*d*x) + 2*a^3*
b^2*d^3*(419*c^2 + 308*c*d*x + 84*d^2*x^2) - 2*a^2*b^3*d^2*(45*c^3 + 262*c^2*d*x
+ 244*c*d^2*x^2 + 72*d^3*x^3) + a*b^4*d*(-65*c^4 + 40*c^3*d*x + 408*c^2*d^2*x^2
+ 416*c*d^3*x^3 + 128*d^4*x^4) + 5*b^5*(15*c^5 - 10*c^4*d*x + 8*c^3*d^2*x^2 + 4
32*c^2*d^3*x^3 + 640*c*d^4*x^4 + 256*d^5*x^5)))/(7680*b^5*d^3) - ((b*c - a*d)^4*
(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d
]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(1024*b^(11/2)*d^(7/2))

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Maple [B]  time = 0.032, size = 1240, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x+c)^(5/2)*(b*x+a)^(1/2),x)

[Out]

-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-336*x^2*a^3*b^2*d^5*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)-80*x^2*b^5*c^3*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^
(1/2)-256*x^4*a*b^4*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-6400*x^4*b^5
*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+288*x^3*a^2*b^3*d^5*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-4320*x^3*b^5*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*(b*d)^(1/2)+180*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^3*d^2*(b*d)^(1
/2)+130*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*d*(b*d)^(1/2)+420*d^5*(b*d*x^2
+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b*(b*d)^(1/2)+100*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*x*b^5*d*(b*d)^(1/2)+1890*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*b*(b*d)^(
1/2)-1676*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*d^3*(b*d)^(1/2)+315*d^6*ln
(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2)
)*a^6+75*c^6*b^6*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))-2560*x^5*b^5*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)-630*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*(b*d)^(1/2)-150*c^5*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*b^5*(b*d)^(1/2)-1050*d^5*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*c*b+1125*c^2*d^4*ln(1/2*(2*b*
d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2-
300*c^3*a^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*
c)/(b*d)^(1/2))*b^3*d^3-75*c^4*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*d^2-90*c^5*a*ln(1/2*(2*b*d*x+2*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^5*d-1232*d^4*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*c*b^2*(b*d)^(1/2)+1048*c^2*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*x*a^2*b^3*d^3*(b*d)^(1/2)-80*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a
*b^4*d^2*(b*d)^(1/2)+976*x^2*a^2*b^3*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)
^(1/2)-816*x^2*a*b^4*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-832*x^3
*a*b^4*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)/b^5/d^3/(b*d)^(1/2)

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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(sqrt(b*x + a)*(d*x + c)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286265, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(4*(1280*b^5*d^5*x^5 + 75*b^5*c^5 - 65*a*b^4*c^4*d - 90*a^2*b^3*c^3*d^2
+ 838*a^3*b^2*c^2*d^3 - 945*a^4*b*c*d^4 + 315*a^5*d^5 + 128*(25*b^5*c*d^4 + a*b
^4*d^5)*x^4 + 16*(135*b^5*c^2*d^3 + 26*a*b^4*c*d^4 - 9*a^2*b^3*d^5)*x^3 + 8*(5*b
^5*c^3*d^2 + 51*a*b^4*c^2*d^3 - 61*a^2*b^3*c*d^4 + 21*a^3*b^2*d^5)*x^2 - 2*(25*b
^5*c^4*d - 20*a*b^4*c^3*d^2 + 262*a^2*b^3*c^2*d^3 - 308*a^3*b^2*c*d^4 + 105*a^4*
b*d^5)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(5*b^6*c^6 - 6*a*b^5*c^5*d
- 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 75*a^4*b^2*c^2*d^4 - 70*a^5*b*c*d^5 +
21*a^6*d^6)*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)*sq
rt(b*d)))/(sqrt(b*d)*b^5*d^3), 1/15360*(2*(1280*b^5*d^5*x^5 + 75*b^5*c^5 - 65*a*
b^4*c^4*d - 90*a^2*b^3*c^3*d^2 + 838*a^3*b^2*c^2*d^3 - 945*a^4*b*c*d^4 + 315*a^5
*d^5 + 128*(25*b^5*c*d^4 + a*b^4*d^5)*x^4 + 16*(135*b^5*c^2*d^3 + 26*a*b^4*c*d^4
- 9*a^2*b^3*d^5)*x^3 + 8*(5*b^5*c^3*d^2 + 51*a*b^4*c^2*d^3 - 61*a^2*b^3*c*d^4 +
21*a^3*b^2*d^5)*x^2 - 2*(25*b^5*c^4*d - 20*a*b^4*c^3*d^2 + 262*a^2*b^3*c^2*d^3
- 308*a^3*b^2*c*d^4 + 105*a^4*b*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) -
15*(5*b^6*c^6 - 6*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 75*a^4
*b^2*c^2*d^4 - 70*a^5*b*c*d^5 + 21*a^6*d^6)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqr
t(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^5*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(x**2*(d*x+c)**(5/2)*(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.332151, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done