3.1460 \(\int (a+b x)^{7/2} \sqrt{c+d x} \, dx\)

Optimal. Leaf size=230 \[ \frac{7 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac{7 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b d^4}+\frac{7 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{192 b d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{240 b d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.359112, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{7 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac{7 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b d^4}+\frac{7 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{192 b d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{240 b d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/2)*Sqrt[c + d*x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.0537, size = 201, normalized size = 0.87 \[ \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}}}{5 d} + \frac{7 \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{40 d^{2}} + \frac{7 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{48 d^{3}} + \frac{7 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 b d^{3}} - \frac{7 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4}}{128 b d^{4}} - \frac{7 \left (a d - b c\right )^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{3}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/2)*(d*x+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.233506, size = 234, normalized size = 1.02 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (105 a^4 d^4+10 a^3 b d^3 (79 c+121 d x)+2 a^2 b^2 d^2 \left (-448 c^2+289 c d x+1052 d^2 x^2\right )+2 a b^3 d \left (245 c^3-161 c^2 d x+128 c d^2 x^2+744 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b d^4}+\frac{7 (b c-a d)^5 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{3/2} d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/2)*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*(79*c + 121*d*x) + 2*a^
2*b^2*d^2*(-448*c^2 + 289*c*d*x + 1052*d^2*x^2) + 2*a*b^3*d*(245*c^3 - 161*c^2*d
*x + 128*c*d^2*x^2 + 744*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*d^4) + (7*(b*c - a*d)^5*Log[b*c + a*d +
2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(256*b^(3/2)*d^(9/2))

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 858, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/2)*(d*x+c)^(1/2),x)

[Out]

1/5/d*(b*x+a)^(7/2)*(d*x+c)^(3/2)+7/40/d*(b*x+a)^(5/2)*(d*x+c)^(3/2)*a+7/48/d*(b
*x+a)^(3/2)*(d*x+c)^(3/2)*a^2+7/64/d*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a^3-7/24/d^2*(b
*x+a)^(3/2)*(d*x+c)^(3/2)*a*b*c-21/64/d^2*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a^2*b*c+21
/64/d^3*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a*b^2*c^2+21/64/d^2*(d*x+c)^(1/2)*(b*x+a)^(1
/2)*a^2*c^2*b-7/32/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^3*b^2-7/256*d/b*((b*x+a)*
(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2
)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^5+35/128/d^2*((b*x+a)*(d*x+c))^
(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*
b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^2*c^3*b^2-35/256/d^3*((b*x+a)*(d*x+c))^(
1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b
+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a*c^4*b^3+35/256*((b*x+a)*(d*x+c))^(1/2)/(d
*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b
*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^4*c-7/32/d*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3*c+7/1
28/d^4*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^4*b^3+7/48/d^3*(b*x+a)^(3/2)*(d*x+c)^(3/2)*
b^2*c^2-7/64/d^4*(b*x+a)^(1/2)*(d*x+c)^(3/2)*b^3*c^3+7/128/b*(d*x+c)^(1/2)*(b*x+
a)^(1/2)*a^4-7/40/d^2*(b*x+a)^(5/2)*(d*x+c)^(3/2)*b*c-35/128/d*((b*x+a)*(d*x+c))
^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2
*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^3*c^2*b+7/256/d^4*((b*x+a)*(d*x+c))^(1/
2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(
a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*c^5*b^4

_______________________________________________________________________________________

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)^(7/2)*sqrt(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.253992, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 490 \, a b^{3} c^{3} d - 896 \, a^{2} b^{2} c^{2} d^{2} + 790 \, a^{3} b c d^{3} + 105 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + 31 \, a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 32 \, a b^{3} c d^{3} - 263 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 161 \, a b^{3} c^{2} d^{2} + 289 \, a^{2} b^{2} c d^{3} + 605 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 105 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b d^{4}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} - 105 \, b^{4} c^{4} + 490 \, a b^{3} c^{3} d - 896 \, a^{2} b^{2} c^{2} d^{2} + 790 \, a^{3} b c d^{3} + 105 \, a^{4} d^{4} + 48 \,{\left (b^{4} c d^{3} + 31 \, a b^{3} d^{4}\right )} x^{3} - 8 \,{\left (7 \, b^{4} c^{2} d^{2} - 32 \, a b^{3} c d^{3} - 263 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{3} d - 161 \, a b^{3} c^{2} d^{2} + 289 \, a^{2} b^{2} c d^{3} + 605 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 105 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)*sqrt(d*x + c),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 - 105*b^4*c^4 + 490*a*b^3*c^3*d - 896*a^2*b^2*c^2*d^
2 + 790*a^3*b*c*d^3 + 105*a^4*d^4 + 48*(b^4*c*d^3 + 31*a*b^3*d^4)*x^3 - 8*(7*b^4
*c^2*d^2 - 32*a*b^3*c*d^3 - 263*a^2*b^2*d^4)*x^2 + 2*(35*b^4*c^3*d - 161*a*b^3*c
^2*d^2 + 289*a^2*b^2*c*d^3 + 605*a^3*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x
+ c) - 105*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 +
5*a^4*b*c*d^4 - 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*d^4), 1/3840*(2*(384*b^4*d^4*x^4 - 105*b^4*c^
4 + 490*a*b^3*c^3*d - 896*a^2*b^2*c^2*d^2 + 790*a^3*b*c*d^3 + 105*a^4*d^4 + 48*(
b^4*c*d^3 + 31*a*b^3*d^4)*x^3 - 8*(7*b^4*c^2*d^2 - 32*a*b^3*c*d^3 - 263*a^2*b^2*
d^4)*x^2 + 2*(35*b^4*c^3*d - 161*a*b^3*c^2*d^2 + 289*a^2*b^2*c*d^3 + 605*a^3*b*d
^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 105*(b^5*c^5 - 5*a*b^4*c^4*d + 1
0*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - 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*
d^4)]

_______________________________________________________________________________________

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((b*x+a)**(7/2)*(d*x+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.308568, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)*sqrt(d*x + c),x, algorithm="giac")

[Out]

Done