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

Optimal. Leaf size=268 \[ \frac{(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]

[Out]

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

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Rubi [A]  time = 0.405382, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.3705, size = 241, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{2}}}{5 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d + 3 b c\right )}{40 b^{2} d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (7 a d + 3 b c\right )}{48 b^{3} d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (7 a d + 3 b c\right )}{64 b^{4} d} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (7 a d + 3 b c\right )}{128 b^{4} d^{2}} + \frac{\left (a d - b c\right )^{4} \left (7 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{9}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.23822, size = 243, normalized size = 0.91 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (34 c+7 d x)-2 a^2 b^2 d^2 \left (173 c^2+111 c d x+28 d^2 x^2\right )+2 a b^3 d \left (30 c^3+109 c^2 d x+88 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^4 d^2}+\frac{(7 a d+3 b c) (b c-a d)^4 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{9/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(34*c + 7*d*x) - 2*a^2
*b^2*d^2*(173*c^2 + 111*c*d*x + 28*d^2*x^2) + 2*a*b^3*d*(30*c^3 + 109*c^2*d*x +
88*c*d^2*x^2 + 24*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x + 744*c^2*d^2*x^2 + 1008*
c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^4*d^2) + ((b*c - a*d)^4*(3*b*c + 7*a*d)*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^(5/2))

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Maple [B]  time = 0.023, size = 942, normalized size = 3.5 \[{\frac{1}{3840\,{b}^{4}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 768\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+2016\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+352\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+1488\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-375\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}+450\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}-150\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d+45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-444\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}+436\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+60\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+680\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}-692\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+120\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-90\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(d*x+c)^(5/2)*(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)+201
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)+352*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)+1488*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^5-375*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+450*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^3-150*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+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^4-444*(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+436*(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+6
0*(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+680*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*a^3*b*c*d^3-692*(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+12
0*(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^2/(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259734, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} - 45 \, b^{4} c^{4} + 60 \, a b^{3} c^{3} d - 346 \, a^{2} b^{2} c^{2} d^{2} + 340 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4} + 48 \,{\left (21 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (93 \, b^{4} c^{2} d^{2} + 22 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (15 \, b^{4} c^{3} d + 109 \, a b^{3} c^{2} d^{2} - 111 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{4} d^{2}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} - 45 \, b^{4} c^{4} + 60 \, a b^{3} c^{3} d - 346 \, a^{2} b^{2} c^{2} d^{2} + 340 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4} + 48 \,{\left (21 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (93 \, b^{4} c^{2} d^{2} + 22 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (15 \, b^{4} c^{3} d + 109 \, a b^{3} c^{2} d^{2} - 111 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{4} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 - 45*b^4*c^4 + 60*a*b^3*c^3*d - 346*a^2*b^2*c^2*d^2
+ 340*a^3*b*c*d^3 - 105*a^4*d^4 + 48*(21*b^4*c*d^3 + a*b^3*d^4)*x^3 + 8*(93*b^4*
c^2*d^2 + 22*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*x^2 + 2*(15*b^4*c^3*d + 109*a*b^3*c^2*
d^2 - 111*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 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*
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)*s
qrt(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^2), 1/3840*(2*(384*b^4*d^4*x^4 - 45*b^4*c^
4 + 60*a*b^3*c^3*d - 346*a^2*b^2*c^2*d^2 + 340*a^3*b*c*d^3 - 105*a^4*d^4 + 48*(2
1*b^4*c*d^3 + a*b^3*d^4)*x^3 + 8*(93*b^4*c^2*d^2 + 22*a*b^3*c*d^3 - 7*a^2*b^2*d^
4)*x^2 + 2*(15*b^4*c^3*d + 109*a*b^3*c^2*d^2 - 111*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 - 5*a*b^4*c^4*d - 10*
a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*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^2)]

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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*(d*x+c)**(5/2)*(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.288925, 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,x, algorithm="giac")

[Out]

Done