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

Optimal. Leaf size=212 \[ -\frac{\sqrt{a} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{3/2}}+\frac{b^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{2 x^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+5 b c)}{4 c x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (a d+11 b c)}{4 c} \]

[Out]

(b*(11*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c) - ((5*b*c + a*d)*(a + b*x)^
(3/2)*Sqrt[c + d*x])/(4*c*x) - ((a + b*x)^(5/2)*Sqrt[c + d*x])/(2*x^2) - (Sqrt[a
]*(15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/(4*c^(3/2)) + (b^(3/2)*(b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[d]

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Rubi [A]  time = 0.654901, antiderivative size = 212, 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 \[ -\frac{\sqrt{a} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{3/2}}+\frac{b^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{2 x^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+5 b c)}{4 c x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (a d+11 b c)}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

(b*(11*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c) - ((5*b*c + a*d)*(a + b*x)^
(3/2)*Sqrt[c + d*x])/(4*c*x) - ((a + b*x)^(5/2)*Sqrt[c + d*x])/(2*x^2) - (Sqrt[a
]*(15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/(4*c^(3/2)) + (b^(3/2)*(b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[d]

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Rubi in Sympy [A]  time = 88.9124, size = 194, normalized size = 0.92 \[ \frac{\sqrt{a} \left (a^{2} d^{2} - 10 a b c d - 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 c^{\frac{3}{2}}} + \frac{b^{\frac{3}{2}} \left (5 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{d}} + \frac{b \sqrt{a + b x} \sqrt{c + d x} \left (a d + 11 b c\right )}{4 c} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{2 x^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + 5 b c\right )}{4 c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(1/2)/x**3,x)

[Out]

sqrt(a)*(a**2*d**2 - 10*a*b*c*d - 15*b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqr
t(a)*sqrt(c + d*x)))/(4*c**(3/2)) + b**(3/2)*(5*a*d + b*c)*atanh(sqrt(d)*sqrt(a
+ b*x)/(sqrt(b)*sqrt(c + d*x)))/sqrt(d) + b*sqrt(a + b*x)*sqrt(c + d*x)*(a*d + 1
1*b*c)/(4*c) - (a + b*x)**(5/2)*sqrt(c + d*x)/(2*x**2) - (a + b*x)**(3/2)*sqrt(c
+ d*x)*(a*d + 5*b*c)/(4*c*x)

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Mathematica [A]  time = 0.636499, size = 236, normalized size = 1.11 \[ -\frac{\sqrt{a} \log (x) \left (a^2 d^2-10 a b c d-15 b^2 c^2\right )}{8 c^{3/2}}+\frac{\sqrt{a} \left (a^2 d^2-10 a b c d-15 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 c^{3/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{a^2}{2 x^2}-\frac{a (a d+9 b c)}{4 c x}+b^2\right )+\frac{b^{3/2} (5 a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

(b^2 - a^2/(2*x^2) - (a*(9*b*c + a*d))/(4*c*x))*Sqrt[a + b*x]*Sqrt[c + d*x] - (S
qrt[a]*(-15*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*Log[x])/(8*c^(3/2)) + (Sqrt[a]*(-15*
b^2*c^2 - 10*a*b*c*d + a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sq
rt[a + b*x]*Sqrt[c + d*x]])/(8*c^(3/2)) + (b^(3/2)*(b*c + 5*a*d)*Log[b*c + a*d +
2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*Sqrt[d])

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Maple [B]  time = 0.022, size = 511, normalized size = 2.4 \[{\frac{1}{8\,c{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{3}{d}^{2}\sqrt{bd}-10\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}bcd\sqrt{bd}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}\sqrt{bd}+20\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{2}cd\sqrt{ac}+4\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{3}{c}^{2}\sqrt{ac}+8\,{x}^{2}{b}^{2}c\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-2\,x{a}^{2}d\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-18\,xabc\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-4\,{a}^{2}c\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^3,x)

[Out]

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c*(ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)+2*a*c)/x)*x^2*a^3*d^2*(b*d)^(1/2)-10*ln((a*d*x+b*c*x+2*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^2*a^2*b*c*d*(b*d)^(1/2)-15*ln((a
*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^2*a*b^2*c^2
*(b*d)^(1/2)+20*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))*x^2*a*b^2*c*d*(a*c)^(1/2)+4*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))*x^2*b^3*c^2*(a*c)^(1/2)+8*x^
2*b^2*c*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-2*x*a^2*d*(b*d)^
(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-18*x*a*b*c*(b*d)^(1/2)*(a*c)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-4*a^2*c*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^2/(a*c)^(1/2)/(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.27456, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*sqrt(d*x + c)/x^3,x, algorithm="fricas")

[Out]

[1/16*(4*(b^2*c^2 + 5*a*b*c*d)*x^2*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b
*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(
b/d) + 8*(b^2*c*d + a*b*d^2)*x) - (15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*x^2*sqrt(a
/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c^2 + (b*c^2 +
a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2)
+ 4*(4*b^2*c*x^2 - 2*a^2*c - (9*a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/
(c*x^2), 1/16*(8*(b^2*c^2 + 5*a*b*c*d)*x^2*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c
+ a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) - (15*b^2*c^2 + 10*a*b*c*d -
a^2*d^2)*x^2*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*
(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2
+ a^2*c*d)*x)/x^2) + 4*(4*b^2*c*x^2 - 2*a^2*c - (9*a*b*c + a^2*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(c*x^2), -1/8*((15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*x^2*sqrt(-
a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*c*sqrt(-a/c
))) - 2*(b^2*c^2 + 5*a*b*c*d)*x^2*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*
c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b
/d) + 8*(b^2*c*d + a*b*d^2)*x) - 2*(4*b^2*c*x^2 - 2*a^2*c - (9*a*b*c + a^2*d)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(c*x^2), -1/8*((15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)
*x^2*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*
c*sqrt(-a/c))) - 4*(b^2*c^2 + 5*a*b*c*d)*x^2*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*
c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) - 2*(4*b^2*c*x^2 - 2*a^2*c
- (9*a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c*x^2)]

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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((b*x+a)**(5/2)*(d*x+c)**(1/2)/x**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.733224, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*sqrt(d*x + c)/x^3,x, algorithm="giac")

[Out]

sage0*x