3.763 \(\int \frac{(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=318 \[ -\frac{\sqrt{c+d x} (63 b c-59 a d) (b c-a d)}{96 a^3 x^2 \sqrt{a+b x}}+\frac{c \sqrt{c+d x} (9 b c-11 a d)}{24 a^2 x^3 \sqrt{a+b x}}-\frac{5 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{11/2} c^{3/2}}+\frac{\sqrt{c+d x} \left (15 a^2 d^2-322 a b c d+315 b^2 c^2\right ) (b c-a d)}{192 a^4 c x \sqrt{a+b x}}+\frac{b \sqrt{c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{192 a^5 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 \sqrt{a+b x}} \]

[Out]

(b*(945*b^3*c^3 - 1785*a*b^2*c^2*d + 839*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c + d*x]
)/(192*a^5*c*Sqrt[a + b*x]) + (c*(9*b*c - 11*a*d)*Sqrt[c + d*x])/(24*a^2*x^3*Sqr
t[a + b*x]) - ((63*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(96*a^3*x^2*Sqrt[a +
b*x]) + ((b*c - a*d)*(315*b^2*c^2 - 322*a*b*c*d + 15*a^2*d^2)*Sqrt[c + d*x])/(1
92*a^4*c*x*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*Sqrt[a + b*x]) - (5*(b*
c - a*d)^2*(63*b^2*c^2 - 14*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(
Sqrt[a]*Sqrt[c + d*x])])/(64*a^(11/2)*c^(3/2))

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Rubi [A]  time = 1.16211, antiderivative size = 318, 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 \[ -\frac{\sqrt{c+d x} (63 b c-59 a d) (b c-a d)}{96 a^3 x^2 \sqrt{a+b x}}+\frac{c \sqrt{c+d x} (9 b c-11 a d)}{24 a^2 x^3 \sqrt{a+b x}}-\frac{5 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{11/2} c^{3/2}}+\frac{\sqrt{c+d x} \left (15 a^2 d^2-322 a b c d+315 b^2 c^2\right ) (b c-a d)}{192 a^4 c x \sqrt{a+b x}}+\frac{b \sqrt{c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{192 a^5 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^5*(a + b*x)^(3/2)),x]

[Out]

(b*(945*b^3*c^3 - 1785*a*b^2*c^2*d + 839*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c + d*x]
)/(192*a^5*c*Sqrt[a + b*x]) + (c*(9*b*c - 11*a*d)*Sqrt[c + d*x])/(24*a^2*x^3*Sqr
t[a + b*x]) - ((63*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(96*a^3*x^2*Sqrt[a +
b*x]) + ((b*c - a*d)*(315*b^2*c^2 - 322*a*b*c*d + 15*a^2*d^2)*Sqrt[c + d*x])/(1
92*a^4*c*x*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*Sqrt[a + b*x]) - (5*(b*
c - a*d)^2*(63*b^2*c^2 - 14*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(
Sqrt[a]*Sqrt[c + d*x])])/(64*a^(11/2)*c^(3/2))

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Rubi in Sympy [A]  time = 95.7557, size = 303, normalized size = 0.95 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{4 a x^{4} \sqrt{a + b x}} - \frac{c \sqrt{c + d x} \left (11 a d - 9 b c\right )}{24 a^{2} x^{3} \sqrt{a + b x}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (59 a d - 63 b c\right )}{96 a^{3} x^{2} \sqrt{a + b x}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (15 a^{2} d^{2} - 322 a b c d + 315 b^{2} c^{2}\right )}{192 a^{4} c x \sqrt{a + b x}} - \frac{b \sqrt{c + d x} \left (15 a^{3} d^{3} - 839 a^{2} b c d^{2} + 1785 a b^{2} c^{2} d - 945 b^{3} c^{3}\right )}{192 a^{5} c \sqrt{a + b x}} + \frac{5 \left (a d - b c\right )^{2} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{11}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**5/(b*x+a)**(3/2),x)

[Out]

-c*(c + d*x)**(3/2)/(4*a*x**4*sqrt(a + b*x)) - c*sqrt(c + d*x)*(11*a*d - 9*b*c)/
(24*a**2*x**3*sqrt(a + b*x)) - sqrt(c + d*x)*(a*d - b*c)*(59*a*d - 63*b*c)/(96*a
**3*x**2*sqrt(a + b*x)) - sqrt(c + d*x)*(a*d - b*c)*(15*a**2*d**2 - 322*a*b*c*d
+ 315*b**2*c**2)/(192*a**4*c*x*sqrt(a + b*x)) - b*sqrt(c + d*x)*(15*a**3*d**3 -
839*a**2*b*c*d**2 + 1785*a*b**2*c**2*d - 945*b**3*c**3)/(192*a**5*c*sqrt(a + b*x
)) + 5*(a*d - b*c)**2*(a**2*d**2 + 14*a*b*c*d - 63*b**2*c**2)*atanh(sqrt(c)*sqrt
(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(11/2)*c**(3/2))

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Mathematica [A]  time = 0.479237, size = 295, normalized size = 0.93 \[ \frac{-15 \log (x) \left (a^2 d^2+14 a b c d-63 b^2 c^2\right ) (b c-a d)^2+15 \left (a^2 d^2+14 a b c d-63 b^2 c^2\right ) (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a} \sqrt{c} \sqrt{c+d x} \left (a^4 \left (-\left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )\right )+a^3 b x \left (72 c^3+244 c^2 d x+337 c d^2 x^2-15 d^3 x^3\right )+a^2 b^2 c x^2 \left (-126 c^2-637 c d x+839 d^2 x^2\right )+105 a b^3 c^2 x^3 (3 c-17 d x)+945 b^4 c^3 x^4\right )}{x^4 \sqrt{a+b x}}}{384 a^{11/2} c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^5*(a + b*x)^(3/2)),x]

[Out]

((2*Sqrt[a]*Sqrt[c]*Sqrt[c + d*x]*(945*b^4*c^3*x^4 + 105*a*b^3*c^2*x^3*(3*c - 17
*d*x) + a^2*b^2*c*x^2*(-126*c^2 - 637*c*d*x + 839*d^2*x^2) + a^3*b*x*(72*c^3 + 2
44*c^2*d*x + 337*c*d^2*x^2 - 15*d^3*x^3) - a^4*(48*c^3 + 136*c^2*d*x + 118*c*d^2
*x^2 + 15*d^3*x^3)))/(x^4*Sqrt[a + b*x]) - 15*(b*c - a*d)^2*(-63*b^2*c^2 + 14*a*
b*c*d + a^2*d^2)*Log[x] + 15*(b*c - a*d)^2*(-63*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]])/(384
*a^(11/2)*c^(3/2))

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Maple [B]  time = 0.051, size = 982, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),x)

[Out]

1/384*(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^4*b*d^4+180*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*c*d^3-1350*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^2*d^2+2100*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^3*d-945*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^4+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^4*a^5*d^4+180*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*c*d^3-1350*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^2*d^2+2100*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^3*d-945*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^4-30*x^4*a^3*b*d^3*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1678*x^4*a^2*b^2*c*d^2*(a*c)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)-3570*x^4*a*b^3*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1890*x^4
*b^4*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^3*a^4*d^3*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+674*x^3*a^3*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-1274*x^3
*a^2*b^2*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+630*x^3*a*b^3*c^3*(a*c)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)-236*x^2*a^4*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4
88*x^2*a^3*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-252*x^2*a^2*b^2*c^3*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)-272*x*a^4*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+144*x*a^3*b*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-96*a^4*c^3*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2))/c/a^5/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)/(b*x+a)^(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((d*x + c)^(5/2)/((b*x + a)^(3/2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74027, 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)^(3/2)*x^5),x, algorithm="fricas")

[Out]

[-1/768*(4*(48*a^4*c^3 - (945*b^4*c^3 - 1785*a*b^3*c^2*d + 839*a^2*b^2*c*d^2 - 1
5*a^3*b*d^3)*x^4 - (315*a*b^3*c^3 - 637*a^2*b^2*c^2*d + 337*a^3*b*c*d^2 - 15*a^4
*d^3)*x^3 + 2*(63*a^2*b^2*c^3 - 122*a^3*b*c^2*d + 59*a^4*c*d^2)*x^2 - 8*(9*a^3*b
*c^3 - 17*a^4*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((63*b^5*c^4
- 140*a*b^4*c^3*d + 90*a^2*b^3*c^2*d^2 - 12*a^3*b^2*c*d^3 - a^4*b*d^4)*x^5 + (63
*a*b^4*c^4 - 140*a^2*b^3*c^3*d + 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*d^4)*
x^4)*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*c*x^5 + a^6*c*x^4)*sqrt(a*c)), -1/384*(2*(48*a^4*c^3 - (945*b
^4*c^3 - 1785*a*b^3*c^2*d + 839*a^2*b^2*c*d^2 - 15*a^3*b*d^3)*x^4 - (315*a*b^3*c
^3 - 637*a^2*b^2*c^2*d + 337*a^3*b*c*d^2 - 15*a^4*d^3)*x^3 + 2*(63*a^2*b^2*c^3 -
122*a^3*b*c^2*d + 59*a^4*c*d^2)*x^2 - 8*(9*a^3*b*c^3 - 17*a^4*c^2*d)*x)*sqrt(-a
*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3
*c^2*d^2 - 12*a^3*b^2*c*d^3 - a^4*b*d^4)*x^5 + (63*a*b^4*c^4 - 140*a^2*b^3*c^3*d
+ 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*d^4)*x^4)*arctan(1/2*(2*a*c + (b*c
+ a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((a^5*b*c*x^5 + a^6*c*x
^4)*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**5/(b*x+a)**(3/2),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)^(3/2)*x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError