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

Optimal. Leaf size=268 \[ \frac{(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^3}{128 b^2 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^2}{192 b^2 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)}{240 b^2 d^2}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (3 a d+7 b c)}{40 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d} \]

[Out]

-((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^2*d^4) + ((b
*c - a*d)^2*(7*b*c + 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(192*b^2*d^3) - ((b*c
- a*d)*(7*b*c + 3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(240*b^2*d^2) - ((7*b*c +
3*a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(40*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(
3/2))/(5*b*d) + ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(
Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(9/2))

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Rubi [A]  time = 0.424705, 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 \[ \frac{(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^3}{128 b^2 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^2}{192 b^2 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)}{240 b^2 d^2}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (3 a d+7 b c)}{40 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^2*d^4) + ((b
*c - a*d)^2*(7*b*c + 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(192*b^2*d^3) - ((b*c
- a*d)*(7*b*c + 3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(240*b^2*d^2) - ((7*b*c +
3*a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(40*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(
3/2))/(5*b*d) + ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(
Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(9/2))

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Rubi in Sympy [A]  time = 47.3731, size = 245, normalized size = 0.91 \[ \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}}}{5 b d} - \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (3 a d + 7 b c\right )}{40 b^{2} d} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (3 a d + 7 b c\right )}{240 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (3 a d + 7 b c\right )}{192 b^{2} d^{3}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (3 a d + 7 b c\right )}{128 b^{2} d^{4}} + \frac{\left (a d - b c\right )^{4} \left (3 a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{5}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(7/2)*(c + d*x)**(3/2)/(5*b*d) - (a + b*x)**(7/2)*sqrt(c + d*x)*(3*a*
d + 7*b*c)/(40*b**2*d) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)*(3*a*d + 7*b
*c)/(240*b**2*d**2) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**2*(3*a*d + 7*b
*c)/(192*b**2*d**3) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3*(3*a*d + 7*b*c)
/(128*b**2*d**4) + (a*d - b*c)**4*(3*a*d + 7*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(s
qrt(b)*sqrt(c + d*x)))/(128*b**(5/2)*d**(9/2))

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Mathematica [A]  time = 0.242287, size = 242, normalized size = 0.9 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (2 c+d x)+2 a^2 b^2 d^2 \left (-173 c^2+109 c d x+372 d^2 x^2\right )+2 a b^3 d \left (170 c^3-111 c^2 d x+88 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^4}+\frac{(3 a d+7 b c) (b c-a d)^4 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{5/2} d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^(5/2)*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*(2*c + d*x) + 2*a^2*b^2
*d^2*(-173*c^2 + 109*c*d*x + 372*d^2*x^2) + 2*a*b^3*d*(170*c^3 - 111*c^2*d*x + 8
8*c*d^2*x^2 + 504*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^2*d^4) + ((b*c - a*d)^4*(7*b*c + 3*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^(5/2)*
d^(9/2))

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Maple [B]  time = 0.021, size = 942, normalized size = 3.5 \[{\frac{1}{3840\,{d}^{4}{b}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 768\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+2016\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+1488\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+352\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}-150\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}+450\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}-375\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+60\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}+436\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}-444\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-90\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+120\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}-692\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+680\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(5/2)*(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)+2016*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)+9
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)+1488*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)-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^5-75*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^4-150*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^3*b^2*c^2*d^3+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
^2*b^3*c^3*d^2-375*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*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+436*(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^3-444*(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+14
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-90*(b*d)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4+120*(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+680
*(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^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((b*x + a)^(5/2)*sqrt(d*x + c)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25899, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 340 \, a b^{3} c^{3} d - 346 \, a^{2} b^{2} c^{2} d^{2} + 60 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + 21 \, a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 22 \, a b^{3} c d^{3} - 93 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 111 \, a b^{3} c^{2} d^{2} + 109 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{2} d^{4}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 340 \, a b^{3} c^{3} d - 346 \, a^{2} b^{2} c^{2} d^{2} + 60 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + 21 \, a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 22 \, a b^{3} c d^{3} - 93 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 111 \, a b^{3} c^{2} d^{2} + 109 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{2} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 - 105*b^4*c^4 + 340*a*b^3*c^3*d - 346*a^2*b^2*c^2*d^
2 + 60*a^3*b*c*d^3 - 45*a^4*d^4 + 48*(b^4*c*d^3 + 21*a*b^3*d^4)*x^3 - 8*(7*b^4*c
^2*d^2 - 22*a*b^3*c*d^3 - 93*a^2*b^2*d^4)*x^2 + 2*(35*b^4*c^3*d - 111*a*b^3*c^2*
d^2 + 109*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 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*
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)*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^2*d^4), 1/3840*(2*(384*b^4*d^4*x^4 - 105*b^4*c
^4 + 340*a*b^3*c^3*d - 346*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 45*a^4*d^4 + 48*(b
^4*c*d^3 + 21*a*b^3*d^4)*x^3 - 8*(7*b^4*c^2*d^2 - 22*a*b^3*c*d^3 - 93*a^2*b^2*d^
4)*x^2 + 2*(35*b^4*c^3*d - 111*a*b^3*c^2*d^2 + 109*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 - 25*a*b^4*c^4*d + 30
*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*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
^2*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*(b*x+a)**(5/2)*(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.30077, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done