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

Optimal. Leaf size=319 \[ -\frac{2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{11/2} \sqrt{d}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{8 b^5}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{12 b^4 (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{3 b^3 (b c-a d)^2}+\frac{4 a (c+d x)^{7/2} (3 b c-5 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2} \]

[Out]

(5*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^5) + (5
*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*b^4*(b*c
- a*d)) + ((b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(
3*b^3*(b*c - a*d)^2) - (2*a^2*(c + d*x)^(7/2))/(3*b^2*(b*c - a*d)*(a + b*x)^(3/2
)) + (4*a*(3*b*c - 5*a*d)*(c + d*x)^(7/2))/(3*b^2*(b*c - a*d)^2*Sqrt[a + b*x]) +
(5*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*
x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(11/2)*Sqrt[d])

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Rubi [A]  time = 0.730366, antiderivative size = 319, 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 \[ -\frac{2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{11/2} \sqrt{d}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{8 b^5}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{12 b^4 (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{3 b^3 (b c-a d)^2}+\frac{4 a (c+d x)^{7/2} (3 b c-5 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^5) + (5
*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*b^4*(b*c
- a*d)) + ((b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(
3*b^3*(b*c - a*d)^2) - (2*a^2*(c + d*x)^(7/2))/(3*b^2*(b*c - a*d)*(a + b*x)^(3/2
)) + (4*a*(3*b*c - 5*a*d)*(c + d*x)^(7/2))/(3*b^2*(b*c - a*d)^2*Sqrt[a + b*x]) +
(5*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*
x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(11/2)*Sqrt[d])

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Rubi in Sympy [A]  time = 66.9065, size = 306, normalized size = 0.96 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{7}{2}}}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{4 a \left (c + d x\right )^{\frac{7}{2}} \left (5 a d - 3 b c\right )}{3 b^{2} \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (21 a^{2} d^{2} - 14 a b c d + b^{2} c^{2}\right )}{3 b^{3} \left (a d - b c\right )^{2}} - \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (21 a^{2} d^{2} - 14 a b c d + b^{2} c^{2}\right )}{12 b^{4} \left (a d - b c\right )} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (21 a^{2} d^{2} - 14 a b c d + b^{2} c^{2}\right )}{8 b^{5}} - \frac{5 \left (a d - b c\right ) \left (21 a^{2} d^{2} - 14 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{11}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*(c + d*x)**(7/2)/(3*b**2*(a + b*x)**(3/2)*(a*d - b*c)) - 4*a*(c + d*x)**(
7/2)*(5*a*d - 3*b*c)/(3*b**2*sqrt(a + b*x)*(a*d - b*c)**2) + sqrt(a + b*x)*(c +
d*x)**(5/2)*(21*a**2*d**2 - 14*a*b*c*d + b**2*c**2)/(3*b**3*(a*d - b*c)**2) - 5*
sqrt(a + b*x)*(c + d*x)**(3/2)*(21*a**2*d**2 - 14*a*b*c*d + b**2*c**2)/(12*b**4*
(a*d - b*c)) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(21*a**2*d**2 - 14*a*b*c*d + b**2*c
**2)/(8*b**5) - 5*(a*d - b*c)*(21*a**2*d**2 - 14*a*b*c*d + b**2*c**2)*atanh(sqrt
(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(8*b**(11/2)*sqrt(d))

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Mathematica [A]  time = 0.333353, size = 214, normalized size = 0.67 \[ \frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{11/2} \sqrt{d}}+\frac{\sqrt{c+d x} \left (315 a^4 d^2+420 a^3 b d (d x-c)+a^2 b^2 \left (113 c^2-574 c d x+63 d^2 x^2\right )-6 a b^3 x \left (-27 c^2+16 c d x+3 d^2 x^2\right )+b^4 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )}{24 b^5 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(315*a^4*d^2 + 420*a^3*b*d*(-c + d*x) - 6*a*b^3*x*(-27*c^2 + 16*c
*d*x + 3*d^2*x^2) + b^4*x^2*(33*c^2 + 26*c*d*x + 8*d^2*x^2) + a^2*b^2*(113*c^2 -
574*c*d*x + 63*d^2*x^2)))/(24*b^5*(a + b*x)^(3/2)) + (5*(b*c - a*d)*(b^2*c^2 -
14*a*b*c*d + 21*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]])/(16*b^(11/2)*Sqrt[d])

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Maple [B]  time = 0.04, size = 1002, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(d*x+c)^(1/2)*(-16*x^4*b^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+315*ln(
1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^3
*b^2*d^3-525*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d
)^(1/2))*x^2*a^2*b^3*c*d^2+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a*b^4*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^5*c^3+36*x^3*a*b^3*d^2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)-52*x^3*b^4*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+63
0*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*
a^4*b*d^3-1050*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b
*d)^(1/2))*x*a^3*b^2*c*d^2+450*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b^3*c^2*d-30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b^4*c^3-126*x^2*a^2*b^2*d^2*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+192*x^2*a*b^3*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)-66*x^2*b^4*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+315*ln(1/2*(2*b*d*x+2*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*d^3-525*ln(1/2*(2*b*d
*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b*c*d^2+225*l
n(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b
^2*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)
^(1/2))*a^2*b^3*c^3-840*x*a^3*b*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1148*x*a
^2*b^2*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-324*x*a*b^3*c^2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)-630*a^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+840*a^3*b*c*d
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-226*a^2*b^2*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(3/2)/b^5

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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((d*x + c)^(5/2)*x^2/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27218, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^4*d^2*x^4 + 113*a^2*b^2*c^2 - 420*a^3*b*c*d + 315*a^4*d^2 + 2*(13*
b^4*c*d - 9*a*b^3*d^2)*x^3 + 3*(11*b^4*c^2 - 32*a*b^3*c*d + 21*a^2*b^2*d^2)*x^2
+ 2*(81*a*b^3*c^2 - 287*a^2*b^2*c*d + 210*a^3*b*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*
sqrt(d*x + c) - 15*(a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 35*a^4*b*c*d^2 - 21*a^5*d^3
+ (b^5*c^3 - 15*a*b^4*c^2*d + 35*a^2*b^3*c*d^2 - 21*a^3*b^2*d^3)*x^2 + 2*(a*b^4
*c^3 - 15*a^2*b^3*c^2*d + 35*a^3*b^2*c*d^2 - 21*a^4*b*d^3)*x)*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)))/((b^7*x^2 + 2*a*b^6*x
+ a^2*b^5)*sqrt(b*d)), 1/48*(2*(8*b^4*d^2*x^4 + 113*a^2*b^2*c^2 - 420*a^3*b*c*d
+ 315*a^4*d^2 + 2*(13*b^4*c*d - 9*a*b^3*d^2)*x^3 + 3*(11*b^4*c^2 - 32*a*b^3*c*d
+ 21*a^2*b^2*d^2)*x^2 + 2*(81*a*b^3*c^2 - 287*a^2*b^2*c*d + 210*a^3*b*d^2)*x)*s
qrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 35*
a^4*b*c*d^2 - 21*a^5*d^3 + (b^5*c^3 - 15*a*b^4*c^2*d + 35*a^2*b^3*c*d^2 - 21*a^3
*b^2*d^3)*x^2 + 2*(a*b^4*c^3 - 15*a^2*b^3*c^2*d + 35*a^3*b^2*c*d^2 - 21*a^4*b*d^
3)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b
*d)))/((b^7*x^2 + 2*a*b^6*x + a^2*b^5)*sqrt(-b*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.668607, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x