3.789 \(\int \frac{(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=278 \[ -\frac{7 b \sqrt{c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac{\sqrt{c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}+\frac{3 c \sqrt{c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}+\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{11/2} \sqrt{c}}-\frac{b \sqrt{c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \]

[Out]

(-7*b*(15*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(24*a^4*(a + b*x)^(3/2)) + (3*
c*(b*c - a*d)*Sqrt[c + d*x])/(4*a^2*x^2*(a + b*x)^(3/2)) - ((21*b*c - 11*a*d)*(b
*c - a*d)*Sqrt[c + d*x])/(8*a^3*x*(a + b*x)^(3/2)) - (b*(315*b^2*c^2 - 420*a*b*c
*d + 113*a^2*d^2)*Sqrt[c + d*x])/(24*a^5*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(3
*a*x^3*(a + b*x)^(3/2)) + (5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(11/2)*Sqrt[c])

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Rubi [A]  time = 1.14685, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{7 b \sqrt{c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac{\sqrt{c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}+\frac{3 c \sqrt{c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}+\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{11/2} \sqrt{c}}-\frac{b \sqrt{c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^(5/2)),x]

[Out]

(-7*b*(15*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(24*a^4*(a + b*x)^(3/2)) + (3*
c*(b*c - a*d)*Sqrt[c + d*x])/(4*a^2*x^2*(a + b*x)^(3/2)) - ((21*b*c - 11*a*d)*(b
*c - a*d)*Sqrt[c + d*x])/(8*a^3*x*(a + b*x)^(3/2)) - (b*(315*b^2*c^2 - 420*a*b*c
*d + 113*a^2*d^2)*Sqrt[c + d*x])/(24*a^5*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(3
*a*x^3*(a + b*x)^(3/2)) + (5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(11/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 160.972, size = 267, normalized size = 0.96 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3 a x^{3} \left (a + b x\right )^{\frac{3}{2}}} - \frac{3 c \sqrt{c + d x} \left (a d - b c\right )}{4 a^{2} x^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (11 a d - 21 b c\right )}{8 a^{3} x \left (a + b x\right )^{\frac{3}{2}}} - \frac{7 b \sqrt{c + d x} \left (a d - b c\right ) \left (7 a d - 15 b c\right )}{24 a^{4} \left (a + b x\right )^{\frac{3}{2}}} - \frac{b \sqrt{c + d x} \left (113 a^{2} d^{2} - 420 a b c d + 315 b^{2} c^{2}\right )}{24 a^{5} \sqrt{a + b x}} - \frac{5 \left (a d - b c\right ) \left (a^{2} d^{2} - 14 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{11}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**4/(b*x+a)**(5/2),x)

[Out]

-c*(c + d*x)**(3/2)/(3*a*x**3*(a + b*x)**(3/2)) - 3*c*sqrt(c + d*x)*(a*d - b*c)/
(4*a**2*x**2*(a + b*x)**(3/2)) - sqrt(c + d*x)*(a*d - b*c)*(11*a*d - 21*b*c)/(8*
a**3*x*(a + b*x)**(3/2)) - 7*b*sqrt(c + d*x)*(a*d - b*c)*(7*a*d - 15*b*c)/(24*a*
*4*(a + b*x)**(3/2)) - b*sqrt(c + d*x)*(113*a**2*d**2 - 420*a*b*c*d + 315*b**2*c
**2)/(24*a**5*sqrt(a + b*x)) - 5*(a*d - b*c)*(a**2*d**2 - 14*a*b*c*d + 21*b**2*c
**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a**(11/2)*sqrt(c))

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Mathematica [A]  time = 0.463397, size = 264, normalized size = 0.95 \[ \frac{\frac{15 \log (x) (a d-b c) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{\sqrt{c}}+\frac{15 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}-\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^4 \left (8 c^2+26 c d x+33 d^2 x^2\right )+6 a^3 b x \left (-3 c^2-16 c d x+27 d^2 x^2\right )+a^2 b^2 x^2 \left (63 c^2-574 c d x+113 d^2 x^2\right )+420 a b^3 c x^3 (c-d x)+315 b^4 c^2 x^4\right )}{x^3 (a+b x)^{3/2}}}{48 a^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^(5/2)),x]

[Out]

((-2*Sqrt[a]*Sqrt[c + d*x]*(315*b^4*c^2*x^4 + 420*a*b^3*c*x^3*(c - d*x) + 6*a^3*
b*x*(-3*c^2 - 16*c*d*x + 27*d^2*x^2) + a^4*(8*c^2 + 26*c*d*x + 33*d^2*x^2) + a^2
*b^2*x^2*(63*c^2 - 574*c*d*x + 113*d^2*x^2)))/(x^3*(a + b*x)^(3/2)) + (15*(-(b*c
) + a*d)*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Log[x])/Sqrt[c] + (15*(b*c - a*d)*(
21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]
*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[c])/(48*a^(11/2))

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Maple [B]  time = 0.048, size = 1009, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^4/(b*x+a)^(5/2),x)

[Out]

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

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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((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56182, 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)/((b*x + a)^(5/2)*x^4),x, algorithm="fricas")

[Out]

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

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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((d*x+c)**(5/2)/x**4/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError