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

Optimal. Leaf size=302 \[ \frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{9/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right )}{64 b^4 d^3}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2-42 b d x (a d+b c)+38 a b c d+35 b^2 c^2\right )}{240 b^3 d^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-7 a^4 d^4-2 a^3 b c d^3+2 a b^3 c^3 d+7 b^4 c^4\right )}{128 b^4 d^4}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d} \]

[Out]

-((7*b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3 - 7*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c +
d*x])/(128*b^4*d^4) - ((b*c + a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*(a + b*x
)^(3/2)*Sqrt[c + d*x])/(64*b^4*d^3) + (x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(5*b
*d) + ((a + b*x)^(3/2)*(c + d*x)^(3/2)*(35*b^2*c^2 + 38*a*b*c*d + 35*a^2*d^2 - 4
2*b*d*(b*c + a*d)*x))/(240*b^3*d^3) + ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*
a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
128*b^(9/2)*d^(9/2))

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Rubi [A]  time = 0.594678, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{9/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right )}{64 b^4 d^3}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2-42 b d x (a d+b c)+38 a b c d+35 b^2 c^2\right )}{240 b^3 d^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-7 a^4 d^4-2 a^3 b c d^3+2 a b^3 c^3 d+7 b^4 c^4\right )}{128 b^4 d^4}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[a + b*x]*Sqrt[c + d*x],x]

[Out]

-((7*b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3 - 7*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c +
d*x])/(128*b^4*d^4) - ((b*c + a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*(a + b*x
)^(3/2)*Sqrt[c + d*x])/(64*b^4*d^3) + (x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(5*b
*d) + ((a + b*x)^(3/2)*(c + d*x)^(3/2)*(35*b^2*c^2 + 38*a*b*c*d + 35*a^2*d^2 - 4
2*b*d*(b*c + a*d)*x))/(240*b^3*d^3) + ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*
a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
128*b^(9/2)*d^(9/2))

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Rubi in Sympy [A]  time = 50.4239, size = 301, normalized size = 1. \[ \frac{x^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{5 b d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{35 a^{2} d^{2}}{4} + \frac{19 a b c d}{2} + \frac{35 b^{2} c^{2}}{4} - \frac{21 b d x \left (a d + b c\right )}{2}\right )}{60 b^{3} d^{3}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + b c\right ) \left (7 a^{2} d^{2} + 2 a b c d + 7 b^{2} c^{2}\right )}{64 b^{4} d^{3}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (7 a^{4} d^{4} + 2 a^{3} b c d^{3} - 2 a b^{3} c^{3} d - 7 b^{4} c^{4}\right )}{128 b^{4} d^{4}} + \frac{\left (a d - b c\right )^{2} \left (a d + b c\right ) \left (7 a^{2} d^{2} + 2 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(a + b*x)**(3/2)*(c + d*x)**(3/2)/(5*b*d) + (a + b*x)**(3/2)*(c + d*x)**(3/
2)*(35*a**2*d**2/4 + 19*a*b*c*d/2 + 35*b**2*c**2/4 - 21*b*d*x*(a*d + b*c)/2)/(60
*b**3*d**3) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d + b*c)*(7*a**2*d**2 + 2*a*b*c*
d + 7*b**2*c**2)/(64*b**4*d**3) + sqrt(a + b*x)*sqrt(c + d*x)*(7*a**4*d**4 + 2*a
**3*b*c*d**3 - 2*a*b**3*c**3*d - 7*b**4*c**4)/(128*b**4*d**4) + (a*d - b*c)**2*(
a*d + b*c)*(7*a**2*d**2 + 2*a*b*c*d + 7*b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(
sqrt(d)*sqrt(a + b*x)))/(128*b**(9/2)*d**(9/2))

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Mathematica [A]  time = 0.25678, size = 264, normalized size = 0.87 \[ \frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{9/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (4 c+7 d x)-2 a^2 b^2 d^2 \left (-17 c^2+11 c d x+28 d^2 x^2\right )+2 a b^3 d \left (20 c^3-11 c^2 d x+8 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^4 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[a + b*x]*Sqrt[c + d*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(4*c + 7*d*x) - 2*a^2*
b^2*d^2*(-17*c^2 + 11*c*d*x + 28*d^2*x^2) + 2*a*b^3*d*(20*c^3 - 11*c^2*d*x + 8*c
*d^2*x^2 + 24*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 + 384*d^4*x^4)))/(1920*b^4*d^4) + ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*
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]*S
qrt[c + d*x]])/(256*b^(9/2)*d^(9/2))

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Maple [B]  time = 0.024, size = 942, normalized size = 3.1 \[{\frac{1}{3840\,{b}^{4}{d}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 768\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+32\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}-30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}-30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-44\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}-44\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}+68\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(1/2)*(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)+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)+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)-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)+32*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)+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^5-75*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^3-30*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^2-75*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+140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*x*a^3*b*d^4-44*(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^3-44*(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-210*(b*d)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*a^4*d^4+80*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*
c*d^3+68*(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+80*(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)/b^4/d^4/(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)*sqrt(d*x + c)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300001, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d + 34 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 11 \, a b^{3} c^{2} d^{2} - 11 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{4} d^{4}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d + 34 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 11 \, a b^{3} c^{2} d^{2} - 11 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{4} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 - 105*b^4*c^4 + 40*a*b^3*c^3*d + 34*a^2*b^2*c^2*d^2
+ 40*a^3*b*c*d^3 - 105*a^4*d^4 + 48*(b^4*c*d^3 + a*b^3*d^4)*x^3 - 8*(7*b^4*c^2*d
^2 - 2*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*x^2 + 2*(35*b^4*c^3*d - 11*a*b^3*c^2*d^2 - 1
1*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*(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)*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^4*d^4), 1/3840*(2*(384*b^4*d^4*x^4 - 105*b^4*c^4 + 40*a*b
^3*c^3*d + 34*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 - 105*a^4*d^4 + 48*(b^4*c*d^3 + a
*b^3*d^4)*x^3 - 8*(7*b^4*c^2*d^2 - 2*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*x^2 + 2*(35*b^
4*c^3*d - 11*a*b^3*c^2*d^2 - 11*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*(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)*arctan(1/2*(2*b*d*x + b*c + a*d)*sq
rt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^4*d^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \sqrt{a + b x} \sqrt{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3*sqrt(a + b*x)*sqrt(c + d*x), x)

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GIAC/XCAS [A]  time = 0.241833, size = 491, normalized size = 1.63 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{13} c d^{7} - 31 \, a b^{12} d^{8}}{b^{15} d^{8}}\right )} - \frac{7 \, b^{14} c^{2} d^{6} + 16 \, a b^{13} c d^{7} - 263 \, a^{2} b^{12} d^{8}}{b^{15} d^{8}}\right )} + \frac{5 \,{\left (7 \, b^{15} c^{3} d^{5} + 9 \, a b^{14} c^{2} d^{6} + 9 \, a^{2} b^{13} c d^{7} - 121 \, a^{3} b^{12} d^{8}\right )}}{b^{15} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, b^{16} c^{4} d^{4} + 2 \, a b^{15} c^{3} d^{5} - 2 \, a^{3} b^{13} c d^{7} - 7 \, a^{4} b^{12} d^{8}\right )}}{b^{15} d^{8}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{4}}\right )}{\left | b \right |}}{1920 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(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^3