3.1470 \(\int \frac{\sqrt{c+d x}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{256 d^4 (c+d x)^{3/2}}{3465 (a+b x)^{3/2} (b c-a d)^5}+\frac{128 d^3 (c+d x)^{3/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{32 d^2 (c+d x)^{3/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{16 d (c+d x)^{3/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (16*d*(c + d*x)^(3/2))/(99*(b*c - a*d)^2*(a + b*x)^(9
/2)) - (32*d^2*(c + d*x)^(3/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (128*d^3*(c + d*x)^(3/2))/(1155*(b*c - a
*d)^4*(a + b*x)^(5/2)) - (256*d^4*(c + d*x)^(3/2))/(3465*(b*c - a*d)^5*(a + b*x)^(3/2))

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Rubi [A]  time = 0.0414111, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{256 d^4 (c+d x)^{3/2}}{3465 (a+b x)^{3/2} (b c-a d)^5}+\frac{128 d^3 (c+d x)^{3/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{32 d^2 (c+d x)^{3/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{16 d (c+d x)^{3/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (16*d*(c + d*x)^(3/2))/(99*(b*c - a*d)^2*(a + b*x)^(9
/2)) - (32*d^2*(c + d*x)^(3/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (128*d^3*(c + d*x)^(3/2))/(1155*(b*c - a
*d)^4*(a + b*x)^(5/2)) - (256*d^4*(c + d*x)^(3/2))/(3465*(b*c - a*d)^5*(a + b*x)^(3/2))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x}}{(a+b x)^{13/2}} \, dx &=-\frac{2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac{(8 d) \int \frac{\sqrt{c+d x}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}+\frac{\left (16 d^2\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac{32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac{\left (64 d^3\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac{2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac{32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac{128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}+\frac{\left (128 d^4\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^{5/2}} \, dx}{1155 (b c-a d)^4}\\ &=-\frac{2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac{32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac{128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac{256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0776076, size = 170, normalized size = 0.99 \[ -\frac{2 (c+d x)^{3/2} \left (198 a^2 b^2 d^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )+924 a^3 b d^3 (2 d x-3 c)+1155 a^4 d^4+44 a b^3 d \left (30 c^2 d x-35 c^3-24 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (240 c^2 d^2 x^2-280 c^3 d x+315 c^4-192 c d^3 x^3+128 d^4 x^4\right )\right )}{3465 (a+b x)^{11/2} (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(3/2)*(1155*a^4*d^4 + 924*a^3*b*d^3*(-3*c + 2*d*x) + 198*a^2*b^2*d^2*(15*c^2 - 12*c*d*x + 8*d^2*
x^2) + 44*a*b^3*d*(-35*c^3 + 30*c^2*d*x - 24*c*d^2*x^2 + 16*d^3*x^3) + b^4*(315*c^4 - 280*c^3*d*x + 240*c^2*d^
2*x^2 - 192*c*d^3*x^3 + 128*d^4*x^4)))/(3465*(b*c - a*d)^5*(a + b*x)^(11/2))

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Maple [A]  time = 0.009, size = 256, normalized size = 1.5 \begin{align*}{\frac{256\,{b}^{4}{d}^{4}{x}^{4}+1408\,a{b}^{3}{d}^{4}{x}^{3}-384\,{b}^{4}c{d}^{3}{x}^{3}+3168\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-2112\,a{b}^{3}c{d}^{3}{x}^{2}+480\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+3696\,{a}^{3}b{d}^{4}x-4752\,{a}^{2}{b}^{2}c{d}^{3}x+2640\,a{b}^{3}{c}^{2}{d}^{2}x-560\,{b}^{4}{c}^{3}dx+2310\,{a}^{4}{d}^{4}-5544\,{a}^{3}bc{d}^{3}+5940\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-3080\,a{b}^{3}{c}^{3}d+630\,{b}^{4}{c}^{4}}{3465\,{a}^{5}{d}^{5}-17325\,{a}^{4}bc{d}^{4}+34650\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-34650\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+17325\,a{b}^{4}{c}^{4}d-3465\,{b}^{5}{c}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(13/2),x)

[Out]

2/3465*(d*x+c)^(3/2)*(128*b^4*d^4*x^4+704*a*b^3*d^4*x^3-192*b^4*c*d^3*x^3+1584*a^2*b^2*d^4*x^2-1056*a*b^3*c*d^
3*x^2+240*b^4*c^2*d^2*x^2+1848*a^3*b*d^4*x-2376*a^2*b^2*c*d^3*x+1320*a*b^3*c^2*d^2*x-280*b^4*c^3*d*x+1155*a^4*
d^4-2772*a^3*b*c*d^3+2970*a^2*b^2*c^2*d^2-1540*a*b^3*c^3*d+315*b^4*c^4)/(b*x+a)^(11/2)/(a^5*d^5-5*a^4*b*c*d^4+
10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 90.3644, size = 1613, normalized size = 9.43 \begin{align*} -\frac{2 \,{\left (128 \, b^{4} d^{5} x^{5} + 315 \, b^{4} c^{5} - 1540 \, a b^{3} c^{4} d + 2970 \, a^{2} b^{2} c^{3} d^{2} - 2772 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} - 64 \,{\left (b^{4} c d^{4} - 11 \, a b^{3} d^{5}\right )} x^{4} + 16 \,{\left (3 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} + 99 \, a^{2} b^{2} d^{5}\right )} x^{3} - 8 \,{\left (5 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} - 231 \, a^{3} b d^{5}\right )} x^{2} +{\left (35 \, b^{4} c^{4} d - 220 \, a b^{3} c^{3} d^{2} + 594 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} + 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3465 \,{\left (a^{6} b^{5} c^{5} - 5 \, a^{7} b^{4} c^{4} d + 10 \, a^{8} b^{3} c^{3} d^{2} - 10 \, a^{9} b^{2} c^{2} d^{3} + 5 \, a^{10} b c d^{4} - a^{11} d^{5} +{\left (b^{11} c^{5} - 5 \, a b^{10} c^{4} d + 10 \, a^{2} b^{9} c^{3} d^{2} - 10 \, a^{3} b^{8} c^{2} d^{3} + 5 \, a^{4} b^{7} c d^{4} - a^{5} b^{6} d^{5}\right )} x^{6} + 6 \,{\left (a b^{10} c^{5} - 5 \, a^{2} b^{9} c^{4} d + 10 \, a^{3} b^{8} c^{3} d^{2} - 10 \, a^{4} b^{7} c^{2} d^{3} + 5 \, a^{5} b^{6} c d^{4} - a^{6} b^{5} d^{5}\right )} x^{5} + 15 \,{\left (a^{2} b^{9} c^{5} - 5 \, a^{3} b^{8} c^{4} d + 10 \, a^{4} b^{7} c^{3} d^{2} - 10 \, a^{5} b^{6} c^{2} d^{3} + 5 \, a^{6} b^{5} c d^{4} - a^{7} b^{4} d^{5}\right )} x^{4} + 20 \,{\left (a^{3} b^{8} c^{5} - 5 \, a^{4} b^{7} c^{4} d + 10 \, a^{5} b^{6} c^{3} d^{2} - 10 \, a^{6} b^{5} c^{2} d^{3} + 5 \, a^{7} b^{4} c d^{4} - a^{8} b^{3} d^{5}\right )} x^{3} + 15 \,{\left (a^{4} b^{7} c^{5} - 5 \, a^{5} b^{6} c^{4} d + 10 \, a^{6} b^{5} c^{3} d^{2} - 10 \, a^{7} b^{4} c^{2} d^{3} + 5 \, a^{8} b^{3} c d^{4} - a^{9} b^{2} d^{5}\right )} x^{2} + 6 \,{\left (a^{5} b^{6} c^{5} - 5 \, a^{6} b^{5} c^{4} d + 10 \, a^{7} b^{4} c^{3} d^{2} - 10 \, a^{8} b^{3} c^{2} d^{3} + 5 \, a^{9} b^{2} c d^{4} - a^{10} b d^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),x, algorithm="fricas")

[Out]

-2/3465*(128*b^4*d^5*x^5 + 315*b^4*c^5 - 1540*a*b^3*c^4*d + 2970*a^2*b^2*c^3*d^2 - 2772*a^3*b*c^2*d^3 + 1155*a
^4*c*d^4 - 64*(b^4*c*d^4 - 11*a*b^3*d^5)*x^4 + 16*(3*b^4*c^2*d^3 - 22*a*b^3*c*d^4 + 99*a^2*b^2*d^5)*x^3 - 8*(5
*b^4*c^3*d^2 - 33*a*b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 - 231*a^3*b*d^5)*x^2 + (35*b^4*c^4*d - 220*a*b^3*c^3*d^2 +
594*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 + 1155*a^4*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^5*c^5 - 5*a^7*b^4*
c^4*d + 10*a^8*b^3*c^3*d^2 - 10*a^9*b^2*c^2*d^3 + 5*a^10*b*c*d^4 - a^11*d^5 + (b^11*c^5 - 5*a*b^10*c^4*d + 10*
a^2*b^9*c^3*d^2 - 10*a^3*b^8*c^2*d^3 + 5*a^4*b^7*c*d^4 - a^5*b^6*d^5)*x^6 + 6*(a*b^10*c^5 - 5*a^2*b^9*c^4*d +
10*a^3*b^8*c^3*d^2 - 10*a^4*b^7*c^2*d^3 + 5*a^5*b^6*c*d^4 - a^6*b^5*d^5)*x^5 + 15*(a^2*b^9*c^5 - 5*a^3*b^8*c^4
*d + 10*a^4*b^7*c^3*d^2 - 10*a^5*b^6*c^2*d^3 + 5*a^6*b^5*c*d^4 - a^7*b^4*d^5)*x^4 + 20*(a^3*b^8*c^5 - 5*a^4*b^
7*c^4*d + 10*a^5*b^6*c^3*d^2 - 10*a^6*b^5*c^2*d^3 + 5*a^7*b^4*c*d^4 - a^8*b^3*d^5)*x^3 + 15*(a^4*b^7*c^5 - 5*a
^5*b^6*c^4*d + 10*a^6*b^5*c^3*d^2 - 10*a^7*b^4*c^2*d^3 + 5*a^8*b^3*c*d^4 - a^9*b^2*d^5)*x^2 + 6*(a^5*b^6*c^5 -
5*a^6*b^5*c^4*d + 10*a^7*b^4*c^3*d^2 - 10*a^8*b^3*c^2*d^3 + 5*a^9*b^2*c*d^4 - a^10*b*d^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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Giac [B]  time = 1.58973, size = 1816, normalized size = 10.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),x, algorithm="giac")

[Out]

-512/3465*(sqrt(b*d)*b^16*c^6*d^5 - 6*sqrt(b*d)*a*b^15*c^5*d^6 + 15*sqrt(b*d)*a^2*b^14*c^4*d^7 - 20*sqrt(b*d)*
a^3*b^13*c^3*d^8 + 15*sqrt(b*d)*a^4*b^12*c^2*d^9 - 6*sqrt(b*d)*a^5*b^11*c*d^10 + sqrt(b*d)*a^6*b^10*d^11 - 11*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^14*c^5*d^5 + 55*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^13*c^4*d^6 - 110*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^12*c^3*d^7 + 110*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^11*c^2*d^8 - 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^2*a^4*b^10*c*d^9 + 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^2*a^5*b^9*d^10 + 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^12*c
^4*d^5 - 220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^11*c^3*d^6 + 330*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^10*c^2*d^7 - 220*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^9*c*d^8 + 55*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^8*d^9 - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^6*b^10*c^3*d^5 + 495*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^6*a*b^9*c^2*d^6 - 495*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^6*a^2*b^8*c*d^7 + 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^7*d
^8 + 330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^8*c^2*d^5 - 660*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^7*c*d^6 + 330*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^6*d^7 + 924*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^6*c*d^5 - 924*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^10*a*b^5*d^6 + 1386*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^12*b^4*d^5)*abs(b)/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^11*b
^2)