3.15.67 \(\int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx\) [1467]

Optimal. Leaf size=66 \[ -\frac {2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {4 d (c+d x)^{3/2}}{15 (b c-a d)^2 (a+b x)^{3/2}} \]

[Out]

-2/5*(d*x+c)^(3/2)/(-a*d+b*c)/(b*x+a)^(5/2)+4/15*d*(d*x+c)^(3/2)/(-a*d+b*c)^2/(b*x+a)^(3/2)

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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \begin {gather*} \frac {4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(5*(b*c - a*d)*(a + b*x)^(5/2)) + (4*d*(c + d*x)^(3/2))/(15*(b*c - a*d)^2*(a + b*x)^(3/2)
)

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]

Rule 47

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])

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}-\frac {(2 d) \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx}{5 (b c-a d)}\\ &=-\frac {2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {4 d (c+d x)^{3/2}}{15 (b c-a d)^2 (a+b x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 46, normalized size = 0.70 \begin {gather*} \frac {2 (c+d x)^{3/2} (-3 b c+5 a d+2 b d x)}{15 (b c-a d)^2 (a+b x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(3/2)*(-3*b*c + 5*a*d + 2*b*d*x))/(15*(b*c - a*d)^2*(a + b*x)^(5/2))

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c + d*x)^(1/2)/(a + b*x)^(7/2),x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(54)=108\).
time = 0.16, size = 128, normalized size = 1.94

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (2 b d x +5 a d -3 b c \right )}{15 \left (b x +a \right )^{\frac {5}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(54\)
default	\(-\frac {\sqrt {d x +c}}{2 b \left (b x +a \right )^{\frac {5}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{4 b}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/b*(d*x+c)^(1/2)/(b*x+a)^(5/2)+1/4*(a*d-b*c)/b*(-2/5*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(5/2)-4/5*d/(-a*d+b*
c)*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(3/2)+4/3*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/(b*x+a)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (54) = 108\).
time = 0.45, size = 175, normalized size = 2.65 \begin {gather*} \frac {2 \, {\left (2 \, b d^{2} x^{2} - 3 \, b c^{2} + 5 \, a c d - {\left (b c d - 5 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} + 3 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

2/15*(2*b*d^2*x^2 - 3*b*c^2 + 5*a*c*d - (b*c*d - 5*a*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^3*b^2*c^2 - 2*a^4*
b*c*d + a^5*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*x^3 + 3*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2
+ 3*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x)/(a + b*x)**(7/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (54) = 108\).
time = 0.06, size = 195, normalized size = 2.95 \begin {gather*} \frac {2 \left (\frac {30 b^{3} d^{6} \sqrt {c+d x} \sqrt {c+d x}}{225 b^{4} c^{2} \left |d\right |-450 b^{3} d a c \left |d\right |+225 b^{2} d^{2} a^{2} \left |d\right |}-\frac {75 b^{3} d^{6} c-75 b^{2} d^{7} a}{225 b^{4} c^{2} \left |d\right |-450 b^{3} d a c \left |d\right |+225 b^{2} d^{2} a^{2} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {a d^{2}-b c d+b d \left (c+d x\right )}}{\left (a d^{2}-b c d+b d \left (c+d x\right )\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(7/2),x)

[Out]

2/15*(2*(d*x + c)*b^3*d^6/(b^4*c^2*abs(d) - 2*a*b^3*c*d*abs(d) + a^2*b^2*d^2*abs(d)) - 5*(b^3*c*d^6 - a*b^2*d^
7)/(b^4*c^2*abs(d) - 2*a*b^3*c*d*abs(d) + a^2*b^2*d^2*abs(d)))*(d*x + c)^(3/2)/((d*x + c)*b*d - b*c*d + a*d^2)
^(5/2)

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Mupad [B]
time = 0.82, size = 127, normalized size = 1.92 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (10\,a\,d^2-2\,b\,c\,d\right )}{15\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {6\,b\,c^2-10\,a\,c\,d}{15\,b^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,d^2\,x^2}{15\,b\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a^2\,\sqrt {a+b\,x}}{b^2}+\frac {2\,a\,x\,\sqrt {a+b\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c + d*x)^(1/2)*((x*(10*a*d^2 - 2*b*c*d))/(15*b^2*(a*d - b*c)^2) - (6*b*c^2 - 10*a*c*d)/(15*b^2*(a*d - b*c)^2
) + (4*d^2*x^2)/(15*b*(a*d - b*c)^2)))/(x^2*(a + b*x)^(1/2) + (a^2*(a + b*x)^(1/2))/b^2 + (2*a*x*(a + b*x)^(1/
2))/b)

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