3.1492 \(\int \frac{(a+b x)^{7/2}}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=183 \[ \frac{35 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 \sqrt{b} d^{9/2}}-\frac{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 d^4}+\frac{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{96 d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d} \]

[Out]

(-35*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*d^4) + (35*(b*c - a*d)^2*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(96*d^3) - (7*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d
*x])/(24*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*d) + (35*(b*c - a*d)^4*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*Sqrt[b]*d^(9/2))

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Rubi [A]  time = 0.249057, antiderivative size = 183, 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 \[ \frac{35 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 \sqrt{b} d^{9/2}}-\frac{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 d^4}+\frac{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{96 d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/2)/Sqrt[c + d*x],x]

[Out]

(-35*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*d^4) + (35*(b*c - a*d)^2*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(96*d^3) - (7*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d
*x])/(24*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*d) + (35*(b*c - a*d)^4*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*Sqrt[b]*d^(9/2))

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Rubi in Sympy [A]  time = 34.3333, size = 165, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x}}{4 d} + \frac{7 \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )}{24 d^{2}} + \frac{35 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}}{96 d^{3}} + \frac{35 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 d^{4}} + \frac{35 \left (a d - b c\right )^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 \sqrt{b} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/2)/(d*x+c)**(1/2),x)

[Out]

(a + b*x)**(7/2)*sqrt(c + d*x)/(4*d) + 7*(a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b
*c)/(24*d**2) + 35*(a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**2/(96*d**3) + 35*
sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3/(64*d**4) + 35*(a*d - b*c)**4*atanh(s
qrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(64*sqrt(b)*d**(9/2))

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Mathematica [A]  time = 0.214678, size = 177, normalized size = 0.97 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (279 a^3 d^3+a^2 b d^2 (326 d x-511 c)+a b^2 d \left (385 c^2-252 c d x+200 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )\right )}{192 d^4}+\frac{35 (b c-a d)^4 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 \sqrt{b} d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/2)/Sqrt[c + d*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(279*a^3*d^3 + a^2*b*d^2*(-511*c + 326*d*x) + a*b^2
*d*(385*c^2 - 252*c*d*x + 200*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x - 56*c*d^2*x
^2 + 48*d^3*x^3)))/(192*d^4) + (35*(b*c - a*d)^4*Log[b*c + a*d + 2*b*d*x + 2*Sqr
t[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(128*Sqrt[b]*d^(9/2))

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Maple [B]  time = 0.011, size = 650, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/2)/(d*x+c)^(1/2),x)

[Out]

1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/d+7/24/d*(b*x+a)^(5/2)*(d*x+c)^(1/2)*a-7/24/d^2*
(b*x+a)^(5/2)*(d*x+c)^(1/2)*b*c+35/96/d*(b*x+a)^(3/2)*(d*x+c)^(1/2)*a^2-35/48/d^
2*(b*x+a)^(3/2)*(d*x+c)^(1/2)*a*b*c+35/96/d^3*(b*x+a)^(3/2)*(d*x+c)^(1/2)*b^2*c^
2+35/64/d*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a^3-105/64/d^2*(b*x+a)^(1/2)*(d*x+c)^(1/2)
*a^2*b*c+105/64/d^3*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a*b^2*c^2-35/64/d^4*(b*x+a)^(1/2
)*(d*x+c)^(1/2)*b^3*c^3+35/128*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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^4-35/32/d*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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*b
*c+105/64/d^2*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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*b^2*c^
2-35/32/d^3*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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*b^3*c^3+35
/128/d^4*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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)*b^4*c^4

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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)^(7/2)/sqrt(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271991, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 385 \, a b^{2} c^{2} d - 511 \, a^{2} b c d^{2} + 279 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 25 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} d - 126 \, a b^{2} c d^{2} + 163 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} d^{4}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 385 \, a b^{2} c^{2} d - 511 \, a^{2} b c d^{2} + 279 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 25 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} d - 126 \, a b^{2} c d^{2} + 163 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/sqrt(d*x + c),x, algorithm="fricas")

[Out]

[1/768*(4*(48*b^3*d^3*x^3 - 105*b^3*c^3 + 385*a*b^2*c^2*d - 511*a^2*b*c*d^2 + 27
9*a^3*d^3 - 8*(7*b^3*c*d^2 - 25*a*b^2*d^3)*x^2 + 2*(35*b^3*c^2*d - 126*a*b^2*c*d
^2 + 163*a^2*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 105*(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)*d^4), 1/384*(2
*(48*b^3*d^3*x^3 - 105*b^3*c^3 + 385*a*b^2*c^2*d - 511*a^2*b*c*d^2 + 279*a^3*d^3
- 8*(7*b^3*c*d^2 - 25*a*b^2*d^3)*x^2 + 2*(35*b^3*c^2*d - 126*a*b^2*c*d^2 + 163*
a^2*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 105*(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)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*d^4)]

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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)**(7/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.248059, size = 362, normalized size = 1.98 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b d} - \frac{7 \,{\left (b c d^{5} - a d^{6}\right )}}{b d^{7}}\right )} + \frac{35 \,{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )}}{b d^{7}}\right )} - \frac{105 \,{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )}}{b d^{7}}\right )} \sqrt{b x + a} - \frac{105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{4}}\right )} b}{192 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/sqrt(d*x + c),x, algorithm="giac")

[Out]

1/192*(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*d) - 7*(b*c*d^5 - a*d^6)/(b*d^7)) + 35*(b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6
)/(b*d^7)) - 105*(b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6)/(b*d^
7))*sqrt(b*x + a) - 105*(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)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a
*b*d)))/(sqrt(b*d)*d^4))*b/abs(b)