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

Optimal. Leaf size=192 \[ -\frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{7/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 b d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{96 b d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{24 b d}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b} \]

[Out]

(5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^3) - (5*(b*c - a*d)^2*(a +
b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d^2) + ((b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*
x])/(24*b*d) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b) - (5*(b*c - a*d)^4*ArcTanh[
(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(7/2))

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Rubi [A]  time = 0.249277, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{7/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 b d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{96 b d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{24 b d}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^3) - (5*(b*c - a*d)^2*(a +
b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d^2) + ((b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*
x])/(24*b*d) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b) - (5*(b*c - a*d)^4*ArcTanh[
(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(7/2))

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Rubi in Sympy [A]  time = 32.8321, size = 167, normalized size = 0.87 \[ \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4 d} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{24 d^{2}} + \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{32 d^{3}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 b d^{3}} - \frac{5 \left (a d - b c\right )^{4} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(5/2)*(c + d*x)**(3/2)/(4*d) + 5*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a
*d - b*c)/(24*d**2) + 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2/(32*d**3)
+ 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3/(64*b*d**3) - 5*(a*d - b*c)**4*at
anh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(64*b**(3/2)*d**(7/2))

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Mathematica [A]  time = 0.161509, size = 180, normalized size = 0.94 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^3 d^3+a^2 b d^2 (73 c+118 d x)+a b^2 d \left (-55 c^2+36 c d x+136 d^2 x^2\right )+b^3 \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d^3}-\frac{5 (b c-a d)^4 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3 + a^2*b*d^2*(73*c + 118*d*x) + a*b^2*d*
(-55*c^2 + 36*c*d*x + 136*d^2*x^2) + b^3*(15*c^3 - 10*c^2*d*x + 8*c*d^2*x^2 + 48
*d^3*x^3)))/(192*b*d^3) - (5*(b*c - a*d)^4*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*S
qrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(128*b^(3/2)*d^(7/2))

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Maple [B]  time = 0., size = 645, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/d*(b*x+a)^(5/2)*(d*x+c)^(3/2)+5/24/d*(b*x+a)^(3/2)*(d*x+c)^(3/2)*a-5/24/d^2*
(b*x+a)^(3/2)*(d*x+c)^(3/2)*b*c+5/32/d*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a^2-5/16/d^2*
(b*x+a)^(1/2)*(d*x+c)^(3/2)*a*b*c+5/32/d^3*(b*x+a)^(1/2)*(d*x+c)^(3/2)*b^2*c^2+5
/64/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3-15/64/d*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2*c+
15/64/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^2*b-5/64/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/
2)*c^3*b^2-5/128*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^4
+5/32*((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-15/64/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^2*c^2*b+5/32/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*c^3*b^2-5/128/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)*c^4*b^3

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250172, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 55 \, a b^{2} c^{2} d + 73 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} + 8 \,{\left (b^{3} c d^{2} + 17 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (5 \, b^{3} c^{2} d - 18 \, a b^{2} c d^{2} - 59 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b d^{3}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 55 \, a b^{2} c^{2} d + 73 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} + 8 \,{\left (b^{3} c d^{2} + 17 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (5 \, b^{3} c^{2} d - 18 \, a b^{2} c d^{2} - 59 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*b^3*d^3*x^3 + 15*b^3*c^3 - 55*a*b^2*c^2*d + 73*a^2*b*c*d^2 + 15*a^
3*d^3 + 8*(b^3*c*d^2 + 17*a*b^2*d^3)*x^2 - 2*(5*b^3*c^2*d - 18*a*b^2*c*d^2 - 59*
a^2*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(b^4*c^4 - 4*a*b^3*c^3*
d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*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^3), 1/384*(2*(48*b^
3*d^3*x^3 + 15*b^3*c^3 - 55*a*b^2*c^2*d + 73*a^2*b*c*d^2 + 15*a^3*d^3 + 8*(b^3*c
*d^2 + 17*a*b^2*d^3)*x^2 - 2*(5*b^3*c^2*d - 18*a*b^2*c*d^2 - 59*a^2*b*d^3)*x)*sq
rt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c
^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(s
qrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b*d^3)]

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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)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.294978, size = 842, normalized size = 4.39 \[ \frac{5 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{3}}\right )}{\left | b \right |} + \frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} a^{2}{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )} a{\left | b \right |}}{b^{2}}}{960 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/960*(5*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x
+ a)/b^2 + (b^7*c*d^5 - 17*a*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^
5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*c^2*d^4 - a^2*b^7*c*d^
5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a
^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sq
rt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*abs(b) + 10*(sqrt(b^2*c +
(b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b^4*d^2) + (b*c*d - a*d^2)/(
b^4*d^4)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sq
rt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^3))*a^2*abs(b)/b^2 + (sqrt(
b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/(b^6*d^2)
+ (b*c*d^3 - 7*a*d^4)/(b^6*d^6)) - 3*(b^2*c^2*d^2 - a^2*d^4)/(b^6*d^6)) - 3*(b^
3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + s
qrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5*d^4))*a*abs(b)/b^2)/b