Optimal. Leaf size=230 \[ -\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{3/2} d^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {52, 65, 223,
212} \begin {gather*} \frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac {7 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 b d^4}+\frac {7 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^3}{192 b d^3}-\frac {7 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{240 b d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int (a+b x)^{7/2} \sqrt {c+d x} \, dx &=\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {(b c-a d) \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx}{10 b}\\ &=\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}-\frac {\left (7 (b c-a d)^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{80 b d}\\ &=-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^3\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{96 b d^2}\\ &=\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}-\frac {\left (7 (b c-a d)^4\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b d^3}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^5\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b d^4}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^2 d^4}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^5\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^2 d^4}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{3/2} d^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.41, size = 166, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 d^4 (a+b x)^4+790 b d^3 (a+b x)^3 (c+d x)-896 b^2 d^2 (a+b x)^2 (c+d x)^2+490 b^3 d (a+b x) (c+d x)^3-105 b^4 (c+d x)^4\right )}{1920 b d^4}+\frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{3/2} d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 239, normalized size = 1.04
method | result | size |
default | \(\frac {\left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}}}{5 d}-\frac {7 \left (-a d +b c \right ) \left (\frac {\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}}}{4 d}-\frac {5 \left (-a d +b c \right ) \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}{3 d}-\frac {\left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}}}{2 d}-\frac {\left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 d}\right )}{2 d}\right )}{8 d}\right )}{10 d}\) | \(239\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 702, normalized size = 3.05 \begin {gather*} \left [-\frac {105 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 490 \, a b^{4} c^{3} d^{2} - 896 \, a^{2} b^{3} c^{2} d^{3} + 790 \, a^{3} b^{2} c d^{4} + 105 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 31 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 32 \, a b^{4} c d^{4} - 263 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 161 \, a b^{4} c^{2} d^{3} + 289 \, a^{2} b^{3} c d^{4} + 605 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{2} d^{5}}, -\frac {105 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 490 \, a b^{4} c^{3} d^{2} - 896 \, a^{2} b^{3} c^{2} d^{3} + 790 \, a^{3} b^{2} c d^{4} + 105 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 31 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 32 \, a b^{4} c d^{4} - 263 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 161 \, a b^{4} c^{2} d^{3} + 289 \, a^{2} b^{3} c d^{4} + 605 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{2} d^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1107 vs.
\(2 (186) = 372\).
time = 0.10, size = 1471, normalized size = 6.40
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,x\right )}^{7/2}\,\sqrt {c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________