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

Optimal. Leaf size=315 \[ -\frac{\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{7/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)}{64 b^4 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2}{128 b^4 d^3}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )}{48 b^3 d^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+5 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d} \]

[Out]

((b*c - a*d)^2*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(128*b^4*d^3) + ((b*c - a*d)*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3/2)
*Sqrt[c + d*x])/(64*b^4*d^2) + ((3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3
/2)*(c + d*x)^(3/2))/(48*b^3*d^2) - ((5*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(
5/2))/(40*b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(5*b*d) - ((b*c - a*d)^
3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*S
qrt[c + d*x])])/(128*b^(9/2)*d^(7/2))

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Rubi [A]  time = 0.676292, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{7/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)}{64 b^4 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2}{128 b^4 d^3}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )}{48 b^3 d^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+5 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(128*b^4*d^3) + ((b*c - a*d)*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3/2)
*Sqrt[c + d*x])/(64*b^4*d^2) + ((3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3
/2)*(c + d*x)^(3/2))/(48*b^3*d^2) - ((5*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(
5/2))/(40*b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(5*b*d) - ((b*c - a*d)^
3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*S
qrt[c + d*x])])/(128*b^(9/2)*d^(7/2))

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Rubi in Sympy [A]  time = 56.6413, size = 301, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{5 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d + 5 b c\right )}{40 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (7 a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{48 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (7 a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{64 b^{4} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (7 a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{128 b^{4} d^{3}} + \frac{\left (a d - b c\right )^{3} \left (7 a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{9}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(d*x+c)**(3/2)*(b*x+a)**(1/2),x)

[Out]

x*(a + b*x)**(3/2)*(c + d*x)**(5/2)/(5*b*d) - (a + b*x)**(3/2)*(c + d*x)**(5/2)*
(7*a*d + 5*b*c)/(40*b**2*d**2) + (a + b*x)**(3/2)*(c + d*x)**(3/2)*(7*a**2*d**2
+ 6*a*b*c*d + 3*b**2*c**2)/(48*b**3*d**2) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d
- b*c)*(7*a**2*d**2 + 6*a*b*c*d + 3*b**2*c**2)/(64*b**4*d**2) + sqrt(a + b*x)*sq
rt(c + d*x)*(a*d - b*c)**2*(7*a**2*d**2 + 6*a*b*c*d + 3*b**2*c**2)/(128*b**4*d**
3) + (a*d - b*c)**3*(7*a**2*d**2 + 6*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(b)*sqrt(c
+ d*x)/(sqrt(d)*sqrt(a + b*x)))/(128*b**(9/2)*d**(7/2))

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Mathematica [A]  time = 0.243342, size = 258, normalized size = 0.82 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (19 c+7 d x)-2 a^2 b^2 d^2 \left (18 c^2+61 c d x+28 d^2 x^2\right )+6 a b^3 d \left (-5 c^3+3 c^2 d x+16 c d^2 x^2+8 d^3 x^3\right )+3 b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )}{1920 b^4 d^3}-\frac{(b c-a d)^3 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{9/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(19*c + 7*d*x) - 2*a^2
*b^2*d^2*(18*c^2 + 61*c*d*x + 28*d^2*x^2) + 6*a*b^3*d*(-5*c^3 + 3*c^2*d*x + 16*c
*d^2*x^2 + 8*d^3*x^3) + 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)))/(1920*b^4*d^3) - ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*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]]
)/(256*b^(9/2)*d^(7/2))

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Maple [B]  time = 0.023, size = 942, normalized size = 3. \[{\frac{1}{3840\,{b}^{4}{d}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 768\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+1056\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+192\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+48\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-225\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}+90\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}+30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}+45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d-45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-244\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}+36\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}-60\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+380\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}-72\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d+90\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x+c)^(3/2)*(b*x+a)^(1/2),x)

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*x^4*b^4*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*(b*d)^(1/2)+96*x^3*a*b^3*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+105
6*x^3*b^4*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-112*x^2*a^2*b^2*d^4*
(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+192*x^2*a*b^3*c*d^3*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*(b*d)^(1/2)+48*x^2*b^4*c^2*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(
b*d)^(1/2)+105*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*d^5-225*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*c*d^4+90*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^3*b^2*c^2*d^3+30*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^3*c^3*d^2+45*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*b^4*c^4*d-45*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*c^5+140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*x*a^3*b*d^4-244*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2
*b^2*c*d^3+36*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^3*c^2*d^2-60*(b*
d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^4*c^3*d-210*(b*d)^(1/2)*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*a^4*d^4+380*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3
*b*c*d^3-72*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^2*d^2-60*(b*d)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d+90*(b*d)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*b^4*c^4)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^4/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)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290469, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} + 45 \, b^{4} c^{4} - 30 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 190 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4} + 48 \,{\left (11 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (3 \, b^{4} c^{2} d^{2} + 12 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (15 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 61 \, a^{2} b^{2} c d^{3} - 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{4} d^{3}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} + 45 \, b^{4} c^{4} - 30 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 190 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4} + 48 \,{\left (11 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (3 \, b^{4} c^{2} d^{2} + 12 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (15 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 61 \, a^{2} b^{2} c d^{3} - 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{4} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 + 45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 +
190*a^3*b*c*d^3 - 105*a^4*d^4 + 48*(11*b^4*c*d^3 + a*b^3*d^4)*x^3 + 8*(3*b^4*c^
2*d^2 + 12*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*x^2 - 2*(15*b^4*c^3*d - 9*a*b^3*c^2*d^2
+ 61*a^2*b^2*c*d^3 - 35*a^3*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15
*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c
*d^4 - 7*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^4*d^3), 1/3840*(2*(384*b^4*d^4*x^4 + 45*b^4*c^4 - 30*
a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4*d^4 + 48*(11*b^4*c*
d^3 + a*b^3*d^4)*x^3 + 8*(3*b^4*c^2*d^2 + 12*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*x^2 -
2*(15*b^4*c^3*d - 9*a*b^3*c^2*d^2 + 61*a^2*b^2*c*d^3 - 35*a^3*b*d^4)*x)*sqrt(-b*
d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d
^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*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^4*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)**(3/2)*(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282521, size = 887, normalized size = 2.82 \[ \frac{\frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{3}}\right )} c{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{13} c d^{7} - 31 \, a b^{12} d^{8}}{b^{15} d^{8}}\right )} - \frac{7 \, b^{14} c^{2} d^{6} + 16 \, a b^{13} c d^{7} - 263 \, a^{2} b^{12} d^{8}}{b^{15} d^{8}}\right )} + \frac{5 \,{\left (7 \, b^{15} c^{3} d^{5} + 9 \, a b^{14} c^{2} d^{6} + 9 \, a^{2} b^{13} c d^{7} - 121 \, a^{3} b^{12} d^{8}\right )}}{b^{15} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, b^{16} c^{4} d^{4} + 2 \, a b^{15} c^{3} d^{5} - 2 \, a^{3} b^{13} c d^{7} - 7 \, a^{4} b^{12} d^{8}\right )}}{b^{15} d^{8}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{4}}\right )} d{\left | b \right |}}{b^{2}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(10*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*
x + a)/b^2 + (b^7*c*d^5 - 17*a*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*
d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*c^2*d^4 - a^2*b^7*c*
d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2
*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) +
sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*c*abs(b)/b^2 + (sqrt(b^
2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^
13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*d^7 - 263*
a^2*b^12*d^8)/(b^15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*
d^7 - 121*a^3*b^12*d^8)/(b^15*d^8))*(b*x + a) - 15*(7*b^16*c^4*d^4 + 2*a*b^15*c^
3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d^8))*sqrt(b*x + a) - 15*(7*b^5
*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7
*a^5*d^5)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
)/(sqrt(b*d)*b^2*d^4))*d*abs(b)/b^2)/b