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

Optimal. Leaf size=315 \[ \frac{\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{9/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right )}{48 b^3 d^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^2}{128 b^3 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)}{192 b^3 d^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d} \]

[Out]

-((b*c - a*d)^2*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(128*b^3*d^4) + ((b*c - a*d)*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2
)*Sqrt[c + d*x])/(192*b^3*d^3) + ((7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*(a + b*x)^
(5/2)*Sqrt[c + d*x])/(48*b^3*d^2) - ((7*b*c + 5*a*d)*(a + b*x)^(5/2)*(c + d*x)^(
3/2))/(40*b^2*d^2) + (x*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(5*b*d) + ((b*c - a*d)^
3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*S
qrt[c + d*x])])/(128*b^(7/2)*d^(9/2))

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Rubi [A]  time = 0.679024, 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 \[ \frac{\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{9/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right )}{48 b^3 d^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^2}{128 b^3 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)}{192 b^3 d^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)^2*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(128*b^3*d^4) + ((b*c - a*d)*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2
)*Sqrt[c + d*x])/(192*b^3*d^3) + ((7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*(a + b*x)^
(5/2)*Sqrt[c + d*x])/(48*b^3*d^2) - ((7*b*c + 5*a*d)*(a + b*x)^(5/2)*(c + d*x)^(
3/2))/(40*b^2*d^2) + (x*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(5*b*d) + ((b*c - a*d)^
3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*S
qrt[c + d*x])])/(128*b^(7/2)*d^(9/2))

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Rubi in Sympy [A]  time = 56.3686, size = 301, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{5 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + 7 b c\right )}{40 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (3 a^{2} d^{2} + 6 a b c d + 7 b^{2} c^{2}\right )}{48 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 6 a b c d + 7 b^{2} c^{2}\right )}{192 b^{3} d^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 6 a b c d + 7 b^{2} c^{2}\right )}{128 b^{3} d^{4}} - \frac{\left (a d - b c\right )^{3} \left (3 a^{2} d^{2} + 6 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{7}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x)**(5/2)*(c + d*x)**(3/2)/(5*b*d) - (a + b*x)**(5/2)*(c + d*x)**(3/2)*
(5*a*d + 7*b*c)/(40*b**2*d**2) + (a + b*x)**(5/2)*sqrt(c + d*x)*(3*a**2*d**2 + 6
*a*b*c*d + 7*b**2*c**2)/(48*b**3*d**2) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b
*c)*(3*a**2*d**2 + 6*a*b*c*d + 7*b**2*c**2)/(192*b**3*d**3) - sqrt(a + b*x)*sqrt
(c + d*x)*(a*d - b*c)**2*(3*a**2*d**2 + 6*a*b*c*d + 7*b**2*c**2)/(128*b**3*d**4)
- (a*d - b*c)**3*(3*a**2*d**2 + 6*a*b*c*d + 7*b**2*c**2)*atanh(sqrt(b)*sqrt(c +
d*x)/(sqrt(d)*sqrt(a + b*x)))/(128*b**(7/2)*d**(9/2))

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Mathematica [A]  time = 0.224984, size = 254, normalized size = 0.81 \[ \frac{\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{7/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (45 a^4 d^4-30 a^3 b d^3 (c+d x)+6 a^2 b^2 d^2 \left (-6 c^2+3 c d x+4 d^2 x^2\right )+2 a b^3 d \left (95 c^3-61 c^2 d x+48 c d^2 x^2+264 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^3 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(45*a^4*d^4 - 30*a^3*b*d^3*(c + d*x) + 6*a^2*b^2*d^
2*(-6*c^2 + 3*c*d*x + 4*d^2*x^2) + 2*a*b^3*d*(95*c^3 - 61*c^2*d*x + 48*c*d^2*x^2
+ 264*d^3*x^3) + b^4*(-105*c^4 + 70*c^3*d*x - 56*c^2*d^2*x^2 + 48*c*d^3*x^3 + 3
84*d^4*x^4)))/(1920*b^3*d^4) + ((b*c - a*d)^3*(7*b^2*c^2 + 6*a*b*c*d + 3*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^(7/2)*d^(9/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(3/2)*(d*x+c)^(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)-1056*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)
-96*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)-48*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)+112*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)+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^5*d^5-45*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^4*b*c*d^4-30*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^3*b^2*c^2*d^3-90*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^2
*b^3*c^3*d^2+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*b^4*c^4*d-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))*b^5*c^5+60*(b*d)^(1/2)*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*x*a^3*b*d^4-36*(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+244*(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^2-140*(
b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^4*c^3*d-90*(b*d)^(1/2)*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*a^4*d^4+60*(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^2-380*(b*d
)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d+210*(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)/d^4/b^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((b*x + a)^(3/2)*sqrt(d*x + c)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2592, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 190 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} - 30 \, a^{3} b c d^{3} + 45 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + 11 \, a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 12 \, a b^{3} c d^{3} - 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 61 \, a b^{3} c^{2} d^{2} + 9 \, a^{2} b^{2} c d^{3} - 15 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{3} d^{4}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 190 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} - 30 \, a^{3} b c d^{3} + 45 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + 11 \, a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 12 \, a b^{3} c d^{3} - 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 61 \, a b^{3} c^{2} d^{2} + 9 \, a^{2} b^{2} c d^{3} - 15 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{3} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*(b*x+a)**(3/2)*(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284303, size = 886, normalized size = 2.81 \[ \frac{\frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{3}}\right )} a{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{13} c d^{7} - 31 \, a b^{12} d^{8}}{b^{15} d^{8}}\right )} - \frac{7 \, b^{14} c^{2} d^{6} + 16 \, a b^{13} c d^{7} - 263 \, a^{2} b^{12} d^{8}}{b^{15} d^{8}}\right )} + \frac{5 \,{\left (7 \, b^{15} c^{3} d^{5} + 9 \, a b^{14} c^{2} d^{6} + 9 \, a^{2} b^{13} c d^{7} - 121 \, a^{3} b^{12} d^{8}\right )}}{b^{15} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, b^{16} c^{4} d^{4} + 2 \, a b^{15} c^{3} d^{5} - 2 \, a^{3} b^{13} c d^{7} - 7 \, a^{4} b^{12} d^{8}\right )}}{b^{15} d^{8}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{4}}\right )}{\left | b \right |}}{b}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*sqrt(d*x + c)*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))*a*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))*abs(b)/b)/b